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THE THEORY AND PRACTICE <A 



\-J 



OF 



zs^'i^ 



SURVEYING. 



DESIGNED FOR THE USE OF 



SURVEYORS AND ENGINEERS GENERALLY, 



BUT ESPECIALLY FOR THE USE OF 



Students in Engineering 



J. B. JOHNSON, C.E., 

Professor of Civil Engineering in ' Washington University, St. Louis, Mo, ; 

Formerly Civil Engineer on the U. S. Lake and Mississippi River 

Surveys ; Member of the American Society of 

Civil Engineers. 

V OF CO 



NEW YORK: 
JOHN WILEY & SONS, 

15 Astor Place. 
1886. 



Copyright, i88t>, 
By J. B. JOHNSON. 



L -S" ?V 1 



PREFACE. 



No apology is necessary for the appearance of a new book 
on Surveying. The needs of surveyors have long been far be- 
yond the accessible literature on this subject, to say nothing of 
that which has heretofore been formulated in text-books. The 
author's object has been to supply this want so far as he was 
able to do it. 

The subject of surveying, both in the books and in the 
schools, has been too largely confined to Land Surveying. The 
engineering graduates of our polytechnic schools are probably 
called upon to do more in any one of the departments of 
Railroad, City, Topographical, Hydrographical, Mining, or 
Geodetic Surveying than in that of Land Surveying. Some 
of these subjects, as for example City, Geodetic, and Hy- 
drographical Surveying, have not been formulated hitherto, 
in any adequate sense, in either English or any other 
language, to the author's knowledge. In the case of Geodetic 
Surveying there has been a wide hiatus between the matter 
given in text-books and the treatment of the subject in works 
on Geodesy and in special reports of geodetic operations. The 
latter were too technical, prolix, and difficult to give to stu- 
dents, while the former was entirely inadequate to any reason- 
able preparation for this kind of work on even a small scale. 
The subjects of City and Hydrographical Surveying as here 
presented are absolutely new. 

Part I. treats of the adjustment, use, and care of all kinds 



IV PREFACE. 

of instruments used in surveying, either in field or office.* In 
describing the adjustments of instruments the object has been 
to present to the mind of the reader the geometrical relations 
from which a rule or method of adjustment would naturally 
follow. The author has no sympathy with descriptions of ad- 
justments as mechanical processes simply to be committed to 
memory, any more than he has with that method of teaching 
geometry wherein the student is allowed to memorize the 
demonstration. 

Many surveying instruments not usually described in books 
on surveying are fully treated, such as planimeters, panto- 
graphs, barometers, protractors, etc. The several sets of prob- 
lems given to be worked out by the aid of the corresponding 
instruments are designed to teach the capacity and limitations 
of such instruments, as well as the more important sources of 
error in their use. This work is such as can be performed 
about the college campus, or in the near vicinity, and is sup- 
posed to be assigned for afternoon or Saturday practice while 
the subject is under consideration by the class. More ex- 
tended surveys require a special field-season for their success- 
ful prosecution.f 

The methods of the differential and integral calculus have 
been sparingly used, as in the derivation of the barometric for- 
mula for elevations, and of the LM Z formulae in Appendix 
D. Such demonstrations may have to be postponed to a later 
period of the course. 



* Certain special appliances, as for example heliotropes, micrometer-screws, 
current-meters, etc., are treated in the subsequent chapters. 

f At Washington University all the engineering Sophomores go into the 
field for four weeks at the end of the college year, and make a general land 
and topographical survey, such as shown in Plate II. At the end of the Junior 
year the civil-engineering students go again for four weeks, making then a 
geodetic and railroad survey. Some distant region is selected where the 
ground, boarding facilities, etc., are suitable. 



PREFACE. V 

Part II. includes descriptions of the theory and practice of 
Surveying Methods in the several departments of Land, Topo- 
graphical, Railroad, Hydrographical, Mining, City, and Geo- 
detic Surveying ; Surveys for the Measurement of Volumes; 
and the Projection of Maps, Map Lettering, and Topographi- 
cal Signs. The author has tried to treat these subjects in a 
concise, scientific, and practical way, giving only the latest and 
most approved methods, and omitting all problems whose 
only claim for attention is that of geometrical interest. 

In treating the trite subject of Land Surveying many prob- 
lems which are more curious than useful have been omitted, 
and several new features introduced. The subjects of com- 
puting areas from the rectangular co-ordinates, and the supply- 
ing of missing data, are made problems in analytical geometry, 
as they should be. A logarithmic Traverse Table for every 
minute of arc from zero to 90 , arranged for all azimuths from 
zero to 360 , to be used in connection with a four-place loga- 
rithmic table, serves to compute the co-ordinates of lines when 
the transit is the instrument used. A traverse table com- 
puted for every 15 minutes of arc is no longer of much value. 
The isogonic declination-curves shown on Plate I. will be found 
to embody all the accessible data up to 1885, and are reduced 
from the U. S. Coast Survey chart. Appendix A will be 
found of great value as outlining the Judicial Functions of the 
Surveyor by the best possible authority. 

The chapter on Mining Surveying was written by Mr. C. 
A. Russell, C.E., U. S. Deputy Mineral Surveyor of Boulder, 
Colorado. He has had an extended experience in Hydro- 
graphic Surveying, in addition to many years' practice in sur- 
veying mines and mining claims. 

The chapter on City Surveying was written by Mr. Wm. 
Bouton, C.E., City Surveyor of St. Louis, Mo. Mr. Bouton 
has done a large proportion of the city surveying in St. Louis 






VI PREFACE. 



for the last twenty years, and has gained an enviable reputa- 
tion as a reliable, scientific, and expert surveyor. 

It is believed that the ripe experience of these gentlemen 
which has been embodied in their respective chapters will ma- 
terially enhance the value of the book. 

The author also desires to acknowledge his indebtedness to 
his friend H. S. Pritchett, Professor of Astronomy in Wash- 
ington University, for valuable assistance in the preparation 
of the matter on Time in Chapter XIV. 

Although the theorems and the notation of the method of 
least squares are not used in this work, yet two problems are 
solved by what is called the method of the arithmetic mean 
(which, when properly defined, is the same as the method of 
least squares), which will serve as a good introduction to the 
study of the method of least squares. These problems are the 
Rating of a Current-meter, in Chapter X., and the Adjustment 
of a Quadrilateral, in Chapter XIV. The author has found 
that such solutions as these serve to make clear to the mind 
of the student exactly what is accomplished by the least- 
square methods of adjusting observations. 

The chapter on Measurement of Volumes is not intended 
to be an exhaustive treatment of the subject of earthwork, but 
certain fundamental theorems and relations are established 
which will enable the student to treat rationally all ordinary 
problems. The particular relation between the Henck pris- 
moid and the warped-surface prismoid is an important one, 
but one which the author had nowhere found. 

An earthwork table (Table X.) has also been prepared 
which gives volumes directly, without correction, for the 
warped-surface prismoid. The author has no knowledge that 
such a table has ever been prepared before. 

A former work by the author on Topographical Surveying 
by the Transit and Stadia is substantially included in this 
book. 



PREFACE. Vll 

The methods recommended for measuring base-lines with 
steel-tapes are new ; but they have been thoroughly tested, 
and are likely to work a material change in geodetic methods. 

The author wishes to acknowledge his obligations to many 
instrument-manufacturers for the privilege they have very 
kindly accorded to him of having electrotype copies made from 
the original plates, for many of the cuts of instruments given 
throughout the book ; persons familiar with the valuable cata- 
logues published by these firms will recognize the makers 
among the following : W. & L. E. Gurley, Troy, N. Y. ; Buff 
& Berger, Boston, Mass. ; Fauth & Co., Washington, D. C. ; 
Queen & Co. and Young & Sons, Philadelphia, Pa. ; Keuffel 
& Esser, New York ; and A. S. Aloe and Blattner & Adam of 
St. Louis, Mo. Also to Mr. W. H. Searles for the privilege 
of using copies of plates from his Field-book for Tables L, 
VI., and VII. 

Hoping this work will assist in lifting the business of sur- 
veying to a higher professional plane, as well as to enlarge its 
boundaries, the author submits it to surveyors and engineers 
generally, but especially to the instructors and students in our 
polytechnic schools, for such crucial tests as the class-room 
only can give. 

J. B. J. 
St. Louis, Sept. 23, 1886. 






TABLE OF CONTENTS. 



PAGE 

Introduction i 

BOOK I. 
CHAPTER I. 

INSTRUMENTS FOR MEASURING DISTANCES. 

The Chain : 

i. The Engineer's Chain . . . . 5 

2. Gunter's Chain 5 

3. Testing the Chain 6 

4. The Use of the Chain 8 

The Steel Tape : 

5. Varieties 9 

6. The Use of Steel Tapes 10 

Exercises with the Chain : 

7-17. Practical Problems n, 12 

CHAPTER II. 
INSTRUMENTS FOR DETERMINING DIRECTIONS. 

The Compass : 

18. The Surveyor's Compass described '. 13 

19. The General Principle of Reversion 15 

20. To make the Plate perpendicular to the Axis of the Socket 16 

21. To make the Plane of the Bubbles perpendicular to the Axis of the 

Socket 16 

22. To adjust the Pivot to the Centre of the Graduated Circle ... 16 



X CONTENTS. 



PAGE 

23. To straighten the Needle 17 

24. To make the Plane of the Sights normal to the Plane of the Bubbles. 17 

25. To make the Diameter through the Zero-graduations lie in the Plane 

of the Sights 17 

26. To remagnetize the Needle 18 

27. The Construction and Use of Verniers 18 

The Declination of the Needle : 

28. The Declination defined 20 

29. The Daily Variation 20 

30. The Secular Variation 21 

31. Isogonic Lines 23 

32. Other Variations of the Declination 29 

33. To find the Declination of the Needle 29 

Use of the Needle Compass : 

34. The Use of the Compass 34 

35. To set off the Declination 36 

36. Local Attractions 36 

37. To establish a Line of a Given Bearing 37 

38. To find the True Bearing of a Line 37 

39. To retrace an Old Line 37 

The Prismatic Compass : 

40. The Prismatic Compass described 38 

Exercises ; . 

41-44. Exercises for the Needle Compass 38, 39 

The Solar Compass : 

45. The Burt Solar Compass 39 

46. Adjustment of the Bubbles 41 

47. Adjustment of the Lines of Collimation 41 

48. Adjustment of the Declination Vernier ,. 42 

49. Adjustment of the Vernier on the Latitude Arc 43 

50. Adjustment of Terrestrial Line of Sight to the Plane of the Polar 

Axis 43 

Use of the Solar Compass ? 

51. Conditions requiring its Use 4< 

52. To find the Declination of the Sun 44 

53. To correct the Declination for Refraction 45 

54. Errors in Azimuth due to Errors in the Declination and Latitude 

Angles 49 

55. Solar Attachments 5 2 

Exercises with the Solar Compass : 

56-^59. Practical Problems 53. 54 



CONTENTS. xi 



CHAPTER III. 
INSTRUMENTS FOR DETERMINING HORIZONTAL LINES. 

PAGE 

Plumb-line and Bubble : 

60. Their Universal Use in Surveying and Astronomical Work 55 

61. The Accurate Measurement of small Vertical Angles 58 

62. The Angular Value of one Division of the Bubble 58 

63. General Considerations 59 

The Engineer's Level : 

64. The Level described 60 

65. Adjustment of Line of Sight and Bubble Axis to Parallel Positions. 63 

66. Lateral Adjustment of Bubble 67 

67. The Wye Adjustment 67 

68. Relative Importance of Adjustments 68 

69. Focussing and Parallax 68 

70. The Levelling-rod. 70 

71. The Use of the Level 71 

Differential Levelling : 

72. Differential Levelling defined 72 

73. Length of Sights 73 

74. Bench-marks 74 

75. The Record 75 

76. The Field-work 76 

Profile Levelling : 

77. Profile Levelling defined 77 

78. The Record 78 

Levelling for Fixing a Grade : 

79. The Work described 81 

The Hand Level: 

80. Locke's Hand Level 81 

Exercises with the Level : 

81-85. Practical Problems 82 

CHAPTER IV. 

INSTRUMENTS FOR MEASURING ANGLES. 

THE TRANSIT. 

86. The Engineer's Transit described 83 

87. The Adjustments stated 86 

88. Adjustment of Plate Bubbles 86 

89. Adjustment of Line of Collimation S7 



Xll CONTENTS. 



PAGE 

go. Adjustment of the Horizontal Axis 87 

91. Adjustment of the Telescope Bubble 89 

92. Adjustment of Vernier on Vertical Circle 89 

93. Relative Importance of Adjustments ... 89 

Instrumental Conditions affecting Accurate Measurements : 

94. Eccentricity of Centres and Verniers 90 

95. Inclination of Vertical Axis 91 

96. Inclination of Horizontal Axis 92 

97. Error in Collimation Adjustment 93 

The Use of the Transit : 

98. To measure a Horizontal Angle 93 

99. To measure a Vertical Angle 94 

100. To run out a Straight Line ... 95 

101. Traversing 97 

The Solar Attachment : 

102. Various Forms described. 99 

103. Adjustments of the Saegmuller Attachment 102 

The Gradienter Attachment : 

104. The Gradienter described 104 

The Care of the Transit : 

105. The Care of the Transit 104 

Exercises with the Transit : 

106-1 14. Practical Problems 105-107 

the sextant. 

115. The Sextant described. 108 

116. The Theory of the Sextant , no 

117. The Adjustment of the Index Glass in 

118. The Adjustment of the Horizon Glass. . . , in 

119. The Adjustment of the Telescope to the Plane of the Sextant in 

120. The Use of the Sextant 112 

Exercises with the Sextant : 

121. i2i#. Practical Problems 112,113 

The Goniograph : 

122. The Double-reflecting Goniograph 113 

CHAPTER V. 
THE PLANE TABLE. 

123. The Plane Table described 117 

124. Adjustment of the Plate Bubbles 119 

125. Adjustment of Horizontal Axis 119 



CONTENTS. Xlll 



PAGE 

126. Adjustment of Vernier and Bubble to Telescopic Line of Sight. . . 119 
The Use of the Plane Table: 

127. General Description of its Use 120 

128. Location by Resection 123 

129. Resection on Three Known Points 123 

130. Resection on Two Known Points 124 

131. The Measurement of the Distances by Stadia 125 

Exercises with the Plane Table : 

132-135. Practical Problems 126 

CHAPTER VI. 

ADDITIONAL INSTRUMENTS USED IN SURVEYING AND PLOTTING. 

The Aneroid Barometer: 

136. The Aneroid described 127 

137. Derivation of Barometric Formulae 129 

138. Use of the Aneroid 136 

The Pedometer : 

139. The Pedometer described , 137 

The Length of Men's Steps 138 

The Odometer : 

140. Description and Use 139 

The Clinometer : 

141. Description and Use 141 

The Optical Square : 

142. Description and Use 142 

The Planimeter : 

143. Description and Use 143 

144. Theory of the Polar Planimeter 144 

145. To find the Length of Arm to use 150 

146. The Suspended Planimeter 152 

147. The Rolling Planimeter 152 

148. Theory of the Rolling Planimeter 154 

149. To Test the Accuracy of a Planimeter 157 

150. The Use of the Planimeter 158 

151. Accuracy of Planimeter Measurements, 160 

The Pantograph : 

152. Description and Theory 161 

Various Styles of Pantographs 163 

153. Use of the Pantograph 165 



XIV CONTENTS. 



PAGE 

Protractors : 

154. Various Styles described 166 

Parallel Rulers : 

155. Description and Use 169 

Scales : 

156. Various Kinds described 169 



BOOK II. 

SUE VE YING ME THODS. 

CHAPTER VII. 

LAND-SURVEYING. 

157. Land-surveying defined 172 

158. Laying out Land 172 

The United States System of Laying out the Public Lands : 

159. Origin and Region of Application of the System 173 

160. The Reference Lines 173 

161. The Division into Townships 174 

162. The Division into Sections 175 

163. The Convergence of the Meridians 176 

164. The Corner Boundaries 178 

Finding the Area of Superficial Contents of Land when the 

Limiting Boundaries are given : 

165. The Area defined 179 

By Triangular Subdivision : 

166. By the Use of the Chain alone 180 

167. By the Use of the Compass or Transit and Chain 180 

168. By the Use of the Transit and Stadia 181 

From Bearing and Length of the Boundary Lines : 

169. The Common Method 181 

170. The Field Notes 182 

171. Method of Computation stated 185 

172. Latitudes, Departures, and Meridian Distances 185 

173. Computation of Latitudes and Departures of the Courses 187 

174. Balancing the Survey 190 

175. The Error of Closure 193 

176. The Form of Reduction 194 

177. Area Correction due to Erroneous Length of Chain 197 



CONTENTS. XV 



PAGE 

Area from the Rectangular Co-ordinates of the Corners : 

178. Conditions of Application of the Method 200 

179. Theory of the Method . 201 

180. The Form of Reduction 203 

Supplying Missing or Erroneous Data : 

181. Equations for Supplying Missing Data — Four Cases 203 

Plotting : 

181a. Plotting the Survey «, 208 

Irregular Areas : 

182. The Method by Offsets at Irregular Intervals 208 

183. The Method by Offsets at Regular Intervals 210 

The Subdivision of Land : 

184. The Problems of Infinite Variety 213 

185. To cut from a Given Tract of Land a Given Area by a Right 

Line starting from a Given Point in the Boundary 213 

186. To cut from a Given Tract of Land a Given Area by a Right Line 

running in a Given Direction 215 

Examples : 

187-196^. Practical Problems 220-222 



CHAPTER VIII. 
TOPOGRAPHICAL SURVEYING £Y THE TRANSIT AND STADIA. 

197. Topographical Survey defined 223 

198. Available Methods 223 

199. Method by Transit and Stadia stated , 224 

Theory of Stadia Measurements : 

200. Fundamental Relations : 224 

201. Method Used on the Government Surveys ' 230 

202. Another Method of Graduating Rods 231 

203. Adaptation of Formulae to Inclined Sights 231 

204. Description and Use of the Stadia Tables 233 

205. Description and Use of the Reduction Diagram 235 

The Instruments': 

206. The Transit '. 235 

207. Setting the Cross-wires 236 

208. Graduating the Stadia Rod , 237 

General Topographical Surveying : 

209. The Topography 241 

210. Methods of Field Work 241 



XVI CONTENTS. 



PAGE 

211. Reduction of the Notes 249 

212. Plotting the Stadia Line 252 

213. Check Readings 253 

214. Plotting the Side Readings 254 

215. Contour Lines 259 

216. The Final Map 262 

217. Topographical Symbols . . . . . 263 

218. Accuracy of the Stadia Method 263 



CHAPTER IX. 
RAILROAD SURVEYING. 

219. Objects of the Survey 265 

220. The Field Work 265 

221. The Maps 267 

222. Plotting the Survey 269 

223. Making the Location on the Map , 271 

224. Another Method 275 

CHAPTER X. 
HYDROGRAPHIC SURVEYING. 

225. Hydrographic Surveying defined , 277 

The Location of Soundings : 

226. Enumeration of Methods 278 

227. By Two Angles read on Shore 279 

228. By Two Angles read in the Boat — The Three-point Problem 279 

229. By one Range and one Angle 282 

230. Buoys, Buoy-flags, and Range-poles 283 

231. By one Range and Time-intervals 284 

232. By means of Intersecting Ranges 284 

233. By Means of Cords or Wires 284 

Making the Soundings : 

234. The Lead 285 

235. The Line 285 

236. Sounding Poles 287 

237. Making Soundings in Running Water 287 

238. The Water-surface Plane of Reference 287 

239. Lines of Equal Depth 288 



CONTENTS. XV11 



PAGE 

240. Soundings on Fixed Cross-sections in Rivers 288 

241. Soundings for the Study of Sand-waves 289 

242. Areas of Cross-section 290 

Bench-marks, Gauges, Water-levels, and Water-Slope : 

243. Bench-marks 291 

244. Water Gauges 291 

245. Water-levels 292 

246. River-slope 293 

The Discharge of Streams : 

247. Measuring Mean Velocities of Water-currents 294 

248. Use of Sub-surface Floats 295 

249. Use of Current Meters , . 300 

250. Rating the Meter 301 

251. Use of Rod Floats 307 

252. Comparison of Methods 308 

253. The Relative Rates of Flow in Different Parts of the Cross section 309 

254. To find the Mean Velocity on the Cross-section 312 

255. Sub-currents 316 

256. The Flow over Weirs 316 

257. Weir Formulae and Corrections 319 

258. The Miner's Inch 322 

259. Formulae for the Flow of Water in Open Channels — Kutter's For- 

mula 323 

260. Cross-sections of Least Resistance 328 

Sediment Observations : 

261. Methods and Objects 329 

262. Collecting the Specimens of Water 331 

263. Measuring out the Samples 331 

264. Siphoning off, Filtering, and Weighing the Sediment 332 

CHAPTER XI, 
MINING SURVEYING. 

265. Definitions » 333 

266. Stations 335 

267. Instruments 335 

268. Mining Claims 339 

Underground Surveys : 

269. Mining Surveying proper 343 

270. To determine the Position of the End or Breast of a Tunnel and 

its Depth below the Surface at that Point 343 



XV111 CONTENTS. 



PAGE 

271. Required, the Distance that a Tunnel will have to be driven to cut 

a Vein with a Certain Dip. — Two Cases 346 

272. Required the Direction and Distance from the Breast of a Tunnel 

to a Shaft, and the Depth at which it'will cut the Shaft 348 

273. To Survey a Mine with its Shafts and Drifts 351 

274. Conclusion 354 



CHAPTER XII. 
CITY SURVEYING. 

275. Land-surveying Methods inadequate in City Work 356 

276. The Transit 357 

277. The Steel Tape 357 

Laying Out a Town Site : 

278. Provision for Growth 359 

279. Contour Maps , / , 360 

280. The Use of Angular Measurements in Subdivisions 360 

281. Laying out the Ground 361 

282. The Plat to be Geometrically consistent 363 

283. Monuments 363 

284. Surveys for Subdivision 365 

285. The Datum-plane 369 

286. The Location of Streets 369 

287. Sewer Systems 370 

288. Water supply 370 

289. The Contour Map 371 

Methods of Measurement : 

290. The Retracing of Lines 371 

291. Erroneous Standards 372 

292. True Standards 373 

293. The Use of the Tape 374 

294. Determination of the " Normal Tension" 376 

295. The Working Tension 380 

296. The Effect of Wind , 381 

297. The Effect of Slope , 382 

298. The Temperature Correction 382 

299. Checks 383 

Miscellaneous Problems : 

300. The Improvement of Streets 384 

301. Permanent Bench-marks 384 



CONTENTS. xix 



PAGE 

302. The Value of an Existing Monument 385 

303. The Significance of Possession 387 

304. Disturbed Corners and Inconsistent Plats l 388 

305. Treatment of Surplus and Deficiency 389 

306. The Investigation and Interpretation of Deeds 391 

307. Office Records 391 

308. Preservation of Lines 392 

309. The Want of Agreement between Surveyors 393 



CHAPTER XIII. 
THE MEASUREMENT OF VOLUMES. 

310. Proposition 394 

311. Grading over Extended Surfaces 396 

312. Approximate Estimates by Means of Contours 399 

313. The Prismoid 402 

314. The Prismoidal Formula 402 

315. Areas of Cross-section 404 

316. The Centre and Side Heights 405 

317. The Area of a Three-level Section 405 

318. Cross-sectioning 406 

319. Three-level Sections, the Upper Surface consisting of two Warped 

Surfaces 408 

320. Construction of Tables for Prismoidal Computation 410 

321. Three-level Sections; the Surface divided into Four Planes by 

Diagonals 413 

322. Comparison of Methods by Diagonals and by Warped Surfaces. . . 415 

323. Preliminary Estimates from the Profiles 417 

324. Borrow-pits 420 

325. Shrinkage of Earthwork 420 

326. Excavations under Water 421 

CHAPTER XIV. 
GEODETIC SURVEYING. 

327. Objects of a Geodetic Survey 424 

328. Triangulation Systems 425 

329. The Base-line and its Connections 427 

330. The Reconnaissance 429 



XX CONTENTS. 



PAGE 

331. Instrumental Outfit for Reconnaissance 431 

332. The Direction of Invisible Stations 432 

333. The Heights of Stations 432 

334. Construction of Stations , 437 

335. Targets 438 

336. Heliotropes 442 

337. Station Marks 444 

Measurement of the Base Line : 

338. Methods 447 

The Steel Tape 449 

339. Method of Mounting and Stretching the Tape 450 

340. M. Jaderin's Method 453 

341. The Absolute Length of Tape 455 

342. The Coefficient of Expansion 456 

343. The Modulus of Elasticity 457 

344. Effect of the Sag '. 457 

345. Temperature Correction 459 

346. Temperature Correction when a Metallic Thermometer is used... 460 

347. Correction for Alignment 462 

348. Correction for Sag 465 

349. Correction for Pull 465 

350. Elimination of Corrections for Sag and Pull 465 

351. To reduce a Broken Base to a Straight Line 468 

352. To reduce the Length of the Base to Sea-level 468 

353. Summary of Corrections 469 

354. To compute any Portion of a Broken Base which cannot be 

directly measured = 472 

355. Accuracy attainable by Steel-tape and Metallic-wire Measure- 

ments , . . . . 473 

Measurement of the Angles : 

356. The Instruments 477 

357. The Filar Micrometer 480 

358. The Programme of Observations 483 

359. The Repeating Method ... , 484 

360. Method by Continuous Reading around the Horizon 485 

361. Atmospheric Conditions 487 

362. Geodetic Night Signals 488 

363. Reduction to the Centre • 488 

Adjustment of the Measured Angles : 

364. Equations of Condition 491 

365. Adjustment of a Triangle 493 



CONTENTS. xxi 



PAGE 

Adjustment of a Quadrilateral : 

366. The Geometrical Conditions 494 

367. The Angle-equation Adjustment 495 

368. The Side-equation Adjustment 497 

369. Rigorous Adjustment for Angle- and Side-equations 501 

Example of Quadrilateral Adjustment 504 

Adjustment of Larger Systems : 

370. Used only in Primary Triangulation 506 

371. Computing the Sides of the Triangles 506 

Latitude and Azimuth : 

372. Conditions ■ 508 

373. Latitude and Azimuth by Observations on Circumpolar Stars at 

Culmination and Elongation 508 

374. The Observation for Latitude 512 

375. First Method 513 

376. Second Method 513 

377. Correction for Observations not on the Meridian 514 

378. The Observation for Azimuth 515 

379. Corrections for Observations near Elongation. 517 

380. The Target 518 

381. The Illumination of Cross-wires 518 

Time and Longitude : 

382. Fundamental Relations 519 

383. Time. , 520 

384. Conversion of a Sidereal into a Mean Solar Time Interval, and vice 

versa 522 

385. To change Mean Time into Sidereal Time 524 

386. To change from Sidereal to Mean Time 525 

387. The Observation for Time 526 

388. Selection of Stars, with List of Southern Time-Stars for each Month. 526 

389. Finding the Mean Time by Transit 530 

390. Finding the Altitude 531 

391. Making the Observations 532 

392. Longitude 534 

393. Computing the Geodetic Positions 535 

394. Example of L M Z Computation 539 

Geodetic Levelling : 

395. Of Two Kinds 54° 

(A) Trigonometrical Levelling : 

396. Refraction .... 540 

397. Formulae for Reciprocal Observations 54 1 



XXli CONTENTS. 



PAGE 

398. Formulae for Observations at One Station only 543 

399. Formulae for an Observed Angle of Depression to a Sea Horizon. . 545 

400. To find the Value of the Coefficient of Refraction 546 

(B) Precise Spirit-Levelling: 

401. Precise Levelling defined 547 

402. The Instruments 548 

403. The Instrumental Constants 550 

404. The Daily Adjustments 553 

405. Field Methods 555 

406. Limits of Error : 558 

407. Adjustment of Polygonal Systems , 559 

408. Determination of the Elevation of Mean Tide 563 



CHAPTER XV. 
PROJECTION OF MAPS, MAP-LETTERING, AND TOPOGRAPHICAL SYMBOLS. 

Projection of Maps: 

409. Purpose of the Map 564 

410. Rectangular Projection 564 

41 1. Trapezoidal Projection 565 

412. The Simple Conic Projection 566 

413. De lTsle's Conic Projection 567 

414. Bonne's Projection 567 

415. The Polyconic Projection 568 

416. Formulae used in the Projection of Maps , 568 

417. Meridian Distances in Table XI 571 

418. Summary 572 

419. The Angle of Convergence of Meridians 574 

Map-Lettering and Topographical Symbols: - j 

420. Map Lettering 575 

421. Topographical Symbols 576 



CONTENTS. xxill 



APPENDIX A. 
The Judicial Functions of Surveyors 579 

APPENDIX B. 
Instructions to U. S. Deputy Mineral Surveyors 589 

APPENDIX C. 
Finite Differences 605 

APPENDIX D. 
Derivation of Geodetic Formulae 611 



TABLES. 



I. — Trigonometrical Formulae 625 

II. — For Converting Metres, Feet, and Chains 629 

III. — Logarithms of Numbers to Four Places 630 

IV. —Logarithmic Traverse Table 632 

V. — Stadia Reductions for Horizontal Distance and for Eleva- 
tion 640 

VI. — Natural Sines and Cosines , 648 

VII. — Natural Tangents and Cotangents 657 

VII. — Values of Coefficient in Kutter's Formula 669 

IX. — Diameters of Circular Conduits, b\ Kutter's Formula 670 

X. — Earthwork Table — Volumes by the Prismoidal Formula 671 

XL— Coordinates for Polyconic Projection 681 



SURVEYING. 



INTRODUCTION. 

Surveying is the art of making such field observations and 
measurements as are necessary to determine positions, areas, 
volumes, or movements on the earth's surface. The field opera- 
tions employed to accomplish any of these ends constitute a 
survey. Accompanying such survey there is usually the field 
record, the computation, and the final maps, plats, profiles, areas, 
or volumes. The art of making all these belongs, therefore, to 
the subject of surveying. v 

Inasmuch as all fixed engineering structures or works involve 
a knowledge of that portion of the earth's surface on which they 
are placed, together with the necessary or resulting changes in 
the same, so the execution of such works is usually accompa- 
nied by the surveys necessary to obtain the required informa- 
tion. Thus surveying is seen to be intimately related to en- 
gineering, but it should not be confounded with it. All 
engineers should have a thorough knowledge of surveying, but 
a surveyor may or may not have much knowledge of engineer- 
ing. 

The subject of Surveying naturally divides itself into — 
I. The Adjustment, Use, and Care of Instruments. 
II. Methods of Field Work. 

III. The Records, Computations, and Final Products. 

All the ordinary instruments that a surveyor may be called 
upon to use in any of the departments of the work will be dis- 
cussed in the following pages. The most approved methods 



INTRODUCTION. 



only will be given for obtaining the desired information, and 
many problems that are more curious than useful will not be 
mentioned. The student is assumed to possess a knowledge of 
geometry, and of plane and spherical trigonometry. He is also 
supposed to be guided by an instructor, and have access to 
most of the instruments here mentioned, with the privilege of 
using them in the field. 

The field work of surveying consists wholly of measuring dis- 
tances, angles, and time, and it is well to remember that no meas- 
urement can ever be made exactly. The first thing the young sur- 
veyor needs to learn, therefore, is the proportionate error allowable 
in the special work assigned him to perform.. It is of the utmost 
importance to his success that he shall thoroughly study this 
subject. He should know what all the sources of error are, and 
their relative importance; also the relative cost of diminishing 
the size of such errors. Then, with a given standard of accuracy, 
he will know how to make the survey of the required standard 
with the least expenditure of time and labor. He must not do 
all parts of the work as accurately as possible, or even with the 
same care. For, if the expense is proportioned to the accuracy 
of results, then he is the most successful surveyor who does his 
work just good enough for the purpose. The relative size of 
the various sources of error is of the utmost importance. One 
should not expend considerable time and labor to reduce the 
error of ?neasurement of a line to i in 10,000 when the unknown 
error in the length of the measuring unit may be as high as 1 
in 1000. 

The surveyor must carefully discriminate, also, between com- 
pensating errors and cumulative errors. A compensating error 
is one which is as likely to be plus as minus, and it is therefore 
largely compensated in, or eliminated from, the result. A 
cumulative error is one which always enters with the same sign, 
and therefore it accumulates in the result. Thus, in chaining, 
the error in setting the pin is a compensating error, while the 
error from erroneous length of chain is a cumulative error. If a 
mile is chained with a 66-foot chain, there are 80 measurements 



INTRODUCTION. 



taken. Suppose the error of setting the pin be 0.5 inch, and the 
error in the length of the chain be 0.1 inch. Now the theory of 
probabilities shows us that in the case of compensating errors 
the square root of the number of errors probably* remains un- 
compensated. The probable error from setting the pins is 
therefore 9 X 0.5 inch = 4.5 inches. The error from erroneous 
length of chain is 80 X o. 1 inch = 8 inches. Thus we see that 
although the error from setting the pins was five times as great as 
that from erroneous length of chain, yet in running one mile, the 
resulting error from the latter cause was nearly twice that from 
the former. A careful study of the various sources of error 
affecting a given kind of work will usually enable the surveyor 
either to greatly add to its accuracy without increasing its cost. 
or to greatly diminish its cost without diminishing its accuracy. 
The surveyor should have no desire except to arrive at the 
truth. This is the true scientific spirit. He should be most 
severely honest with himself. He should not allow himself 
to change or " fudge" his notes without sufficient warrant, 
and then a full explanation should be made in his note-book. 
Neither should he make his results appear more accurate than 
they really are.. He should always know, what was about the 
relative accuracy with which his field work was done, and carry 
his results only so far as the accuracy of the work would war- 
rant. He is either foolish or dishonest who, having made a 
survey of an area, for instance, with an error of closure of 1 in 
300, should carry his results to six significant figures, thus giv- 
ing the area to 1 in 500,000. It is usual to carry the computa- 
tions one place farther than the results are known, in order that 
no additional error may come in from the computation. It is 
not unusual, however, to see results given in published docu- 
ments to two, three, or even four places farther than the observa- 
tions would warrant. 



*The meaning of this statement is that on the average this will occur oftener 
than any other combination, and that any single result will, on the average, be 
nearer to this result than to any other. 



INTRODUCTION. 



The student should make himself familiar with the structure 
and use of every part of every instrument put into his hands. 
The best way of doing this is to take the instrument all apart 
and put it together again. This, of course, is not practicable for 
each student in college, but when he is given an instrument in 
real practice, he should then make himself thoroughly familiar 
with it before attempting to use it. 

The adjustments of instruments should be studied as problems 
in descriptive geometry and not as mechanical manipulations, 
learned in a mechanical way; and when adjusting an instrument 
the geometry of the problem should be in the mind rather than 
the rule in the memory. 

Students of engineering in polytechnic schools are urged to 
make themselves familiar with every kind of instrument in the 
outfit of the institution, and to do in the field every kind of work 
herein described if possible. Otherwise he may be called upon 
to do, or to direct others to do, what he has never done himself, 
and he will then find that his studies prove of little avail with- 
out the real knowledge that comes only from experience. 



BOOK I 

ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 



CHAPTER I. 
INSTRUMENTS FOR MEASURING DISTANCES. 

THE CHAIN. 

1. The Engineer's Chain is 50 or 100 feet long, atid 
should be made of No. 12 steel wire. The links are one foot 
long, including the connecting rings. All joints in rings 
and links should be brazed to prevent 
giving. The connections are designed 
so as to admit of as little stretch as 
possible. Every tenth foot is marked 
by a special form of brass tag. If the 
chain is adjustable in length, it should 
be made of standard length by meas- 
uring from the inside of the handle at 
one end to the outside of the handle 
at the other. If it is not adjustable, 
measure from the outside of the handle 
at the rear end to the standard mark 
at the forward end. 

2. Gunter's Chain is 66 feet long, and is divided into 100 
links, each link being 7.92 inches in length. This chain is 
mostly used in land-surveying, where the acre is the unit of 
measure. It was invented by Edmund Gunter, an English 




6 SURVEYING. 



astronomer, about 1620, and is very convenient for obtaining 
areas in acres or distances in miles. Thus, 
One mile = 80 chains ; also, 
One acre = 160 square rods, 
= 10 square chains, 
= 100,000 square links. 
If, therefore, the unit of measure be chains and hundredths 
(links), the area is obtained in square chains and decimals, and 
by pointing off one more place the result is obtained in acres. 
This is the length of chain used on all the U. S. land surveys. 
In all deeds of conveyance and other documents, when the 
word chain is used it is Gunter's chain that is meant. 

3. Testing the Chain. — No chain, of whatever material 
or manufacture, will remain of constant length. The length 
changes from temperature, wear, and various kinds of distor- 
tion. A change of temperature of 70 F. in a 100-foot chain 
will change its length by 0.05 foot, or a change of I in 2000. 

If the links of a chain are joined by three rings, then there 
are eight wearing surfaces for each link, or eight hundred 
wearing surfaces for a 66- or 100-foot chain. If each surface 
should wear 0.01 inch, the chain is lengthened by eight inches. 
It is not uncommon for a railroad survey of, say, 300 miles to 
be run with a single chain. If such a chain were of exactly the 
right length at the beginning of the survey, it might be six 
inches too long at the end of it. 

The change of length from distortion may come from a 
flattening out of the connecting rings, from bending the links, 
or from stretching the chain beyond its elastic limit, thus giv- 
ing it a permanent set. Both the wear and the distortion are 
likely to be less for a steel chain than for an iron one. When 
a bent link is straightened it is permanently lengthened. 

When we remember that all unknown changes in the 
length of the chain produce cumulative errors in the meas- 
ured lines, we see how important it is that the true length of 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 7 

the chain should be always known, or better, that the standard 
length (50, 66, or 100 feet) should be properly measured from 
one end of the chain and marked at the other. This chain 
test is most readily accomplished by the aid of a standard steel 
tape, which is at least as long as the chain. By the aid of such 
a tape a standard length may be laid off on the floor of a large 
room, or two stones may be firmly set in the ground at the 
proper distance apart and marks cut upon their upper sur- 
faces. If stones are used they should reach below the frost- 
line. Or a short tape, or other standard measuring unit, may 
be used for laying off such a base-line. By whatever means it 
is accomplished, some ready means should at all times be 
available for testing the chain. Since a chain always grows 
longer with use, the forward end of the chain will move 
farther and farther from the standard mark. A small file- 
mark may be made on the handle or elsewhere, and then re- 
moved when a new test gives a new position. Care must be 
exercised to see that there are no kinks in the chain either in 
testing or in use. 

In laying out the standard base the temperature at which 
the unit of measure is standard should be known (this tempera- 
ture is stamped on the better class of steel tapes), and if the 
base is not laid out at this temperature, a correction should 
be made before the marks are set. The coefficient of expansion 
of iron and steel is very nearly 0.0000065 for i° F. If T be 
the temperature at which the tape is standard, T the tem- 
perature at which the base is measured, and L the length 
of the base, then 0.0000065 (7^— T)L is the correction to be 
applied to the measured length to give the true length. 

When the chain is tested by this standard base the tem- 
perature should be again noted, and if this is about the mean 
temperature for the field measurements no correction need be 
made to the field work. If it is known, at the time the chain 
is tested, that the temperature is very different from the prob- 



8 SURVEYING. 



able mean of the field work, then the standard mark can be so 
placed on the chain as to make it standard when in use. 

4. The Use of the Chain.— The chain is folded by taking 
it by the middle joint and folding the two ends simultaneous- 
ly. It is opened by taking the two handles in one hand and 
throwing the chain out with the other. 

Since horizontal distances are always desired in surveying, 
the chain should be held horizontally in measuring. Points 
vertically below the ends of the chain are marked by iron pins, 
the head chainman placing them and the rear chainman remov- 
ing them after the next pin is set. The chain is lined in either 
by the head or rear chainman, or by the observer at the instru- 
ment, according as the range-pole is in the rear, or in front, or 
not visible by either chainman. When chaining on level 
ground, the rear chainman brings the outside of the handle 
against the pin, and the head chainman sets the forward side 
of his pin even with the standard mark on the chain. By this 
means the centres of the pins are the true distance apart. On 
uneven ground both chainmen cannot hold to the pin ; one end 
being elevated in order to bring the chain to a horizontal 
position. In this case there are three difficulties to be over- 
come. The chain should be drawn so taut that the stretch 
from the pull would balance the shortening from the sag; the 
chain should be made horizontal ; the elevated end-mark must 
be transferred vertically to the ground. It is practically im- 
possible to do any of these exactly. The first could be deter- 
mined by trial. Stretch the chain between two points at the 
same elevation, having it supported its entire length. Then 
remove the supports, and see how strong a pull is required to 
bring it to the marks again. This should be done by the chain- 
men themselves, thus enabling them to judge how hard to pull 
it when it is off the ground. To hold the chain horizontal on 
sloping ground is very difficult, on account of the judgment 
being usually very much in error as to the position of a hori- 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 9 

zontal line. In all such cases the apparently horizontal line is 
much too nearly parallel with the ground. Sometimes a level has 
been attached to one end of the chain, in which case it should 
be adjusted to indicate horizontal end-positions for a certain 
pull, this being the pull necessary to overcome the shortening 
from sag. To hold a plumb-line at the proper mark, with the 
chain at the right elevation, and stretched the proper amount, 
requires a steady hand in order that the plumb-bob may hang 
stationary. This should be near the ground, and when all is 
ready, it is dropped by the chainman letting go the string. 
The pin is then stuck and the work proceeds. It is common 
in this country for the rear chainman to call " stick" when he 
is ready, and for the head chainman to answer " stuck" when 
he has set the pin. The rear chainman then pulls his pin and 
walks on. 

There should be eleven pins, marked with strips of colored 
flannel tied in the rings to assist in. finding them in grass or 
brush. In starting, the rear chainman takes a pin for the initial 
point, leaving the head chainman with ten pins. When the 
last pin is stuck, the head chainman calls " out," and waits by 
this station until the rear chainman comes up and delivers over 
the ten pins now in his possession. The eleventh pin is in the 
ground, and serves as the initial point for the second score. 
Thus only every ten chains need be scored. 

Good chaining, therefore, consists in knowing the length of 
the chain, in true alignment, horizontal and vertical, and in 
proper stretching, marking, and scoring. 

THE STEEL TAPE. 

5. Varieties. — Steel tapes are now made from one yard to 
1000 feet in length, graduated metrically, or in feet and tenths. 
A pocket steel tape from three to ten feet long should always 
be carried by the surveyor. A 50-foot tape is best fitted to 
city surveying where there are appreciable grades. For cities 



IO SURVEYING. 



without grades a ioo-foot tape might be found more useful. 
For measuring base-lines, or for some kinds of mining surveying, 
a 300 or 500 foot tape is best. These are of small cross-section, 
being about 0.1 inch wide and 0.02 inch thick. A tape about 




Fig. 2. 

0.5 inch wide and 0.02 inch thick (Fig. 2) is perhaps best suited 
to general surveying. 

6. The Use of Steel Tapes. — Steel tape-measures are used 
just as chains are. They are provided with handles, but the 
end graduation-marks are usually on the tape itself and not on 
the handle. They are graduated to order, the graduations 
being either etched or made on brass sleeves which are fastened 
on the tape. Their advantages are many. They do not kink, 
stretch, or wear so as to change their length, so that, with 
careful handling, they remain of constant length except for 
temperature. They are used almost exclusively in city and 
bridge work, and in the measurement of secondary base-lines. 
The same precautions must be taken in regard to alignment, 
pull, and marking with the tape, as was described for the 
chain.* 

* For methods of using the steel tape in accurate measurements, see Chap- 
ter XIIL, Base-Line Measurements. 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. II 



EXERCISES. 

To be worked out on the ground by the use of the chain or tape alone. 

7. To chain a line over a hill between two given points, not visible from 
each other. 

Range-poles are set at the given points. Then the two chainmen, each with 
a range-pole, range themselves in between the two fixed points, near the top 
of the hill, by successive approximations. The line can then be chained. 

8. To chain a line across a valley between two fixed points. 

Establish other range-poles by means of a plumb-line held on range between 
the points. 

9. To chain a line between two fixed points when woods intervene, and the 
true line is not to be cleared out. 

Range out a trial line by poles, leaving fixed points. Find the resulting error 
at the terminus, and move all the points over their proportionate amount. The 
true line may then be chained. 

10. To set a stake in a line perpendicular to a given line at a given point. 
All multiples of 3, 4, and 5 are the sides of a right-angled triangle; also any 

angle in a semicircumference is a right angle. 

11. To find where a perpendicular from a given point without a line will meet 
that line. 

Run an inclined line from the given point to the given line. Erect a per- 
pendicular from the given line near the required point, extend it till it intersects 
the inclined line, and solve by similar triangles. 

12. To establish a second point that shall make with a given point a line 
parallel to a given line. 

Diagonals of a parallelogram bisect each other. 

13. To determine the horizontal distance from a given point to a visible but 
inaccessible object. 

Use two similar right-angled triangles. 

14. To prolong a line beyond an obstacle* in azimuth* and distance. 
First Solution : By an equilateral triangle. 

Second Solution : By two rectangular offsets on each side of the obstacle. 
Third Solution : By similar triangles, as in Fig. 3. 

From any point as A run the line AB, fixing the half and three quarter points 
at x and y. From any other point as C, run CxD, making xD = Cx. From D 

*The azimuth of a line is the angle it forms with the meridian, and is meas- 
ured from the south point in the direction S.W. N.E. to 360 degrees. It thus 
becomes a definite direction when the angle alone is given. Thus the azimuth 
of 220 corresponds to the compass-bearing of N. 40 E. 



12 



SURVEYING. 



run DyE making DE — AB= <\Dy, fixing the middle point z. From B run 
BzH, making zH = Bz. Then is i/ii parallel and equal to DB, AC, and CH. 



A 


C 








E 




X 






H 

z 








-*<r- 







Fig. 3. 

Stakes should be set at all the points lettered in the figure. Check: Measure 
HE and AC. If they are equal the work is correct. 

15. To measure a given angle. 

Lay off equal distances, b, from the vertex on the two lines, and measure the 

a 
third side a of the triangle. Then tan $ A= — ■. 

y ^b 11 — a* 

16. To lay out a given angle on the ground. 

Reverse the above operation. A is known; assume b and compute a. Then 
from A measure off AB — b. From B and A strike arcs with radii equal to a 
and b respectively, giving an intersection at C Then CAB is the required 
angle. If b is assumed not greater than 0.6 the length of the chain, angles may 
be laid out up to 90 . 



17. Other Instruments for measuring distances with great 
accuracy will be discussed under the head of Base-Line 
Measurements, Chapter XIV. 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 1 3 



CHAPTER II. 
INSTRUMENTS FOR DETERMINING DIRECTIONS. 

THE COMPASS. 

18. The Surveyor's Compass consists essentially of a line 
of sight attached to a horizontal graduated circle, at the centre 
of which is suspended a magnetic needle free to move, the 
whole conveniently supported with devices for levelling. Fig. 




Fig. 4. 

4 shows a very "good form of such an instrument. In ad- 
dition to the above essential features, the instrument here 
shown has a tangent-screw and vernier-scale at e for setting 
off the declination of the needle; a tangent-scale on the edge 
of the vertical sight for reading vertical angles, the eye being 
placed at the sight-disk shown on the opposite standard ; and an 



1 4' SURVEYING. 



auxiliary graduated circle, with vernier, shown on the front 
part of the plate, for reading angles closer than could be done 
with the needle. The compass is mounted either on a tripod 
or on a single support called a Jacob's-staff. It is connected 
to its support by a ball-and-socket joint, which furnishes a con- 
venient means of levelling. 

Although the needle-compass does not give very accurate 
results, it is one of the most useful of surveying instruments. 
Its great utility lies in the fact that the needle always points 
in a known direction, and therefore the direction of any line 
of sight may be determined by referring it to the needle-bear- 
ing. The needle points north in only a few localities ; but its 
declination from the north point is readily determined for any 
region, and then the true azimuth, or bearing of a line, may be 
found. It has grown to be the universal custom, in finding 
the direction of a line by the compass, to refer it to either 
the north or the south point, according to which one gives an 
acute angle. Thus, if the bearing is ioo° from the south 
point it is but 8o° from the north point, and the direction 
would be defined as north, 8o° east or west, as the case 
might be: thus no line can have a numerical bearing of 
more than 90 . In accordance with this custom, all needle- 
compasses are graduated from both north and south points 
each way to the east and west points, the north and south 
points being marked zero, and the east and west points 90 . 
When the direction of a line is given by this system it is 
called the bearing of the line. When it is simply referred 
to the position of the needle it is called the magnetic bearing. 
When it is corrected for the declination of the needle, 
either by setting off the declination on the declination-arc or 
by correcting the observed reading, it is called the true bear- 
ing, being then referred to the true meridian. 

Because the graduated circle is attached to the line of sight 
and moves with it, while the needle remains stationary, E and 



ADJUSTMENT, USE, AND CARE OE INSTRUMENTS. 15 

W are placed on the compass-circle in reversed position. 
Thus when the line of sight is north-east, the north end of the 
needle points to the left of the north point on the circle, and 
hence E must be put on this side of the meridian line. 

In reading the compass, always keep the north end of the circle 
pointing forward along the line, and read the north end of the 
needle. 

The north end of the needle is usually shaped to a special 
design, or, if not, it may be distinguished by knowing that the 
south end is weighted by having a small adjustable brass wire 
slipped upon it to overcome the tendency the north end has 
to dip. 

ADJUSTMENTS OF THE COMPASS. 

19. The General Principle of almost all instrumental ad- 
justments is the Principle of Reversion, whereby the error is 
doubled and at the same time made apparent. A thorough mas- 
tery of this principle will nearly always enable one to deter- 
mine the proper method of adjusting all parts of any survey- 
ing instrument. It should be a recognized principle in sur- 
veying, that no one is competent to handle any instrument 
who is not able to determine when it is in exact adjustment, 
to locate the source of the error if not in adjustment, to dis- 
cuss the effect of any error of adjustment on the work in 
hand, and to properly adjust all the movable parts. The 
methods of adjustment should not be committed to memory — 
any more than should the demonstration of a proposition in 
geometry. The student in reading the methods of adjust- 
ment should see that they are correct, just as he sees the cor- 
rectness of a geometrical demonstration. Having thus had 
the method and the reason therefor clearly in the mind, he 
should trust his ability to evolve it again whenever called 
upon. He thus relies upon the accuracy of his reasoning, 
rather than on the distinctness of his recollection. 



1 6 SURVEYING. 



20. To make the Plate perpendicular to the Axis of the 
Socket. — This must be done by the maker. It is here men- 
tioned because the axis is so likely to get accidentally bent. 
Instruments made of soft brass must be handled very care- 
fully to prevent such an accident. If this adjustment is found 
to be very much out, it should be sent to the makers. If 
much out, it will be shown by the needle, and also by the 
plate-bubbles. 

21. To make the Plane of the Bubbles perpendicular 
to the Axis of the Socket. — Level it in one position, turn 
i8o°, and correct one half the movement of each bubble by 
the adjusting-screw at the end of the bubble-case. Now level 
up again, and revolve i8o°, and the bubbles should remain at 
the centre. If not, adjust for one half the movement again, 
and so continue until the bubbles remain in the centre for all 
positions of the plate. 

The student should construct a figure to illustrate this and almost all other 
adjustments. Thus, in this case, let the figure consist of two lines, one repre- 
senting the axis of the socket, and the other the axis of the bubble, crossing it. 
Now if these two lines are not at right angles to each other, when the one is 
horizontal (as the bubble-axis is when the bubble rests at the centre of its tube) 
the other is inclined from the vertical. Now with this latter fixed, let the 
figure be revolved i8o° about it (or construct another figure representing such 
a movement), and it will be seen that the bubble-axis now deviates from the 
horizontal by twice the difference between the angle of the lines and 90 . By 
now correcting one half of this change of direction on the part of the bubble- 
axis, it will be made perpendicular to the socket-axis. Then by relevelling the 
instrument, which consists of moving the socket-axis until the bubbles return 
to the middle of the tubes, the instrument should now revolve in a horizontal 
plane. 

22. To adjust the Pivot to the Centre of the Graduated 
Circle. — When the two ends of the needle do not read exactly 
alike it may be due to one or more of three causes : The 
circle may not be uniformly graduated ; the pivot maybe bent 
out of its central position ; or the needle may be bent. All 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. IJ 

our modern instruments are graduated by machinery, so that 
they have no errors of graduation that could be detected by 
eye. One or both of the other two causes must therefore ex- 
ist. If the difference between the two end-readings is con- 
stant for all positions of the needle, then the pivot is in the 
centre of the circle, but the needle is bent. If the difference 
between the two end-readings is variable for different parts of 
the circle, then the pivot is bent, and the needle may or may 
not be straight. To adjust the pivot, therefore, find the posi- 
tion of the needle which gives the maximum difference of end- 
readings, remove the needle, and bend the pivot at right angles 
to this position by one half the difference in the extreme variation 
of end-readings. Repeat the test, etc. Since the glass cover 
is removed from the compass-box in making this adjustment, 
it should be made indoors, to prevent any disturbance from 
wind. 

23. To straighten the Needle, set the north end exactly 
at some graduation-mark, and read the south end. If not 180 
apart, bend the needle until they are. This implies that the 
preceding adjustment has been made, or examined and found 
correct. 

24. To make the Plane of the Sights normal to the 
Plane of the Bubbles. — Carefully level the instrument and 
bring the plane of the sights upon a suspended plumb-line. 
If this seems to traverse the farther slit, then that sight is in 
adjustment. Reverse the compass, and test the other sight 
in like manner. If either be in error, its base must be re- 
shaped to make it vertical. 

25. To make the Diameter through the Zero-gradua- 
tions lie in the Plane of the Sights. — This should be done by 
the maker, but it can be tested by stretching two fine hairs 
vertically in the centres of the slits. The two hairs and the 
two zero-graduations should then be seen to lie in the same 
plane. The declination-arc must be set to read zero. 

2 



1 8 SURVEYING. 



26. To remagnetize the Needle. — Needles sometimes lose 
their magnetic properties. They must then be remagnetized. 
To do this take a simple bar-magnet and rub each end of the 
needle, from centre towards the ends, with the end of the 
magnet which attracts in each case. In returning the magnet 
for the next stroke lift it up a foot or so to remove it from 
the immediate magnetic field, otherwise it would tend to nul- 
lify its own action. The needle should be removed from the 
pivot in this operation, and the work continued until it shows 
due activity when suspended. An apparently sluggish needle 
may be due to a blunt pivot. If so, this should be removed, 
and ground down on an oil-stone. 

THE VERNIER. 

27. The Vernier is an auxiliary scale used for reading frac- 
tional parts of the divisions on .the main graduated scale or limb. 
If we wish to read to tenths of one division on the limb, we 
make 10 divisions on the vernier correspond to either 9 or 11 
divisions on the limb. Then each division on the vernier is 
one tenth less or greater than a division on the limb. If we 
wish to read to twentieths or thirtieths of one division on the 
limb, there must- be twenty or thirty divisions on the vernier 
corresponding to one more or less on the limb. 

The zero of the vernier-scale marks the point on the limb 
whose reading is desired. 

Suppose this zero-point corresponds exactly with a division 
on the limb. The reading is then made wholly on the limb. 
If a division on the vernier is less than a division on the limb, 
then, by moving the vernier forward a trifle, the next forward 
division on the vernier corresponds with a division on the limb. 
(The particular division on the limb that may be in coincidence 
is of no consequence.) On the other hand, if a division on the 
vernier is greater tha'n a division on the limb, then by moving 
the vernier forward a trifle, the next backward division on the 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 1 9 



vernier comes into coincidence. Thus we have two kinds of 
verniers, called direct and retrograde according as they are read 
forward or backward from the zero-point. Most verniers in 
use are of the direct kind, but those commonly found on sur- 
veyors' compasses for setting off the declination are generally 
of the retrograde order. 

i'O * 




Fig. 5. 

In Fig. 5 are shown two direct verniers, such as are used 
on transits with double graduations. Thus in reading to the 
right the reading is 138 45', but in reading to the left it is 22 1° 
15'. In each case we look along the vernier in the direction of 
the graduation for the coincident lines. 




Fig. 6. 



In Fig. 6 is shown a special form of retrograde vernier in 
which the same set of graduation-lines on the vernier serve for 



20 SURVEYING. 



either right- or left-hand angles. Here a division of the vernier 
is larger than a division on the limb, and it must therefore be 
read backwards. Thus, we see that the zero of the vernier 
is to the left of the zero of the limb, the angle being 30' and 
something more. Starting now toward the right (backwards) 
on the vernier scale, we reach the end or 15-minute mark, 
without finding coincident lines ; we then skip to the left-hand 
side of the vernier scale and proceed towards the right again 
until we find coincident lines at the twenty-sixth mark. The 
reading is therefore 30-1-26=56 minutes. This is the form 
of vernier usually found on surveyors' compasses for setting 
off the declination. We have therefore the following 

* 

Rules. 

First. To find the " smallest reading' of the vernier, divide 
the valtie of a division on the limb by the number of divisions in 
the vernier, 

Second. Read forward along the limb to the last graduation 
preceding the zero of the vernier ; then read forzvard along the 
vernier if direct, or backward if retrograde \ until coincident lines 
are found. The number of this line on the vernier from the zero- 
graduation is the number of " smallest-reading" units to be 
added to the reading made on the limb. 

These rules apply to all verniers, whether linear or circular. 

THE DECLINATION OF THE NEEDLE. 

28. The Declination* of the Needle is the horizontal 
angle it makes with the true meridian. At no place on the 
earth is this angle a constant. The change in this angle is 
called the variation of the declination. 

20. The Daily Variation in the Declination consists in a 

* Formerly called variation of the needle, and still so called by navigators 
and by many surveyors. 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 



21 



swinging of the needle through an arc of about eight minutes 
daily, the north end having its extreme easterly variation about 
8 A.M. and its extreme westerly position about 1. 30 P.M. It 
has its mean or true declination about 10.30 A.M. and 8 P.M. 
It varies with the latitude and with the season, but the follow- 
ing table gives a fair average for the United States. A more 
extended table may be found in the Report of the U. S. Coast 
and Geodetic Survey for 1881, Appendix 8. 

TABLE OF CORRECTIONS TO REDUCE OBSERVED BEARINGS 

TO THE DAILY MEAN. 



Month. 


Add to N.E. and S.W. 

bearings. 

Subtract from N.W. and 

S.E. bearings. 


Add to N.W. and S.E. bearings. 
Subtract from N.E. and S.W. bearings. 


6 

A.M. 


7 

A.M. 


8 

A.M. 


9 
A.M. 


10 
A.M. 


11 

A.M. 


12 

M. 


1 

P.M. 


2 
P.M. 


3 

P.M. 


4 

P.M. 


5 

P.M. 


6 

P.M. 




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I' 


7' 


2' 


i' 


o' 


2' 


3' 
5 
5 
3 


3' 
5 
5 

3 


2' 


i' 


i' 


o' 




3 
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1 


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5 
2 


4 

5 

2 


3 
4 
2 


I 


1 


4 
4 

3 


4 
4 
2 


3 
3 

1 






July 


I 








October 


I 


1 


O 


O 





This table is correct to the nearest minute for Philadelphia, where the observations were 

made. 

30. The Secular Variation of the magnetic declination is 
probably of a periodic character, requiring two or three cen- 
turies to complete a cycle. The most extensive set of obser- 
vations bearing on this subject have been made at Paris, where 
records of the magnetic declination have been kept for about 
three and a half centuries. The secular variation for Paris is 
shown in Fig. 7, and that for Baltimore, Md., in Fig. 8.* 

Whether or not either of these curves will return in time to 
the same extreme limits here given is unknown, as is also the 
cause of these remarkable changes. The extraordinary varia- 
tion in the declination at Paris of some 32 , and that at 



These taken from the Coast Survey Report of 1882. 



22 



SURVEYING. 



Baltimore of some 5 , show the necessity of paying careful 
attention to this matter. No reliance should be placed on 



West 23°222l201918171615141312 1110 9 87654321012345678 910111 


1540 
60 
80 

1600 
20 
40 
60 
80 

1700 
20 
40 
60 
80 

1800 
20 
40 
60 
'80 

1900J 




1 | | [ | | | | | 1 1 


II 1 1 1 1 II I 
























Secular Vdriation\ of the Magnetic Decimation, at Pai 


is 


France.. 


























Observed declinations are shown by dots. 








































Computed \declinations, by first periodic 








































term of formula, by the 


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f* 
































































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Fig. 7. 

Secular Variation of the Magnetic Declination Baltimore,Md. 
West 6° 5 4 3 2 1 r 




Fig. 8. 



old determinations of the declination unless the rate of change 
be known, and even then this rate is not likely to be constant 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 2$ 

a great many years. They also show the necessity of record- 
ing the date and the declination of the needle on all plats and 
records of surveys, with a note stating whether the bearings 
given were the true or magnetic bearings at the time they 
were taken. 

31. Isotonic Lines are imaginary lines on the earth's sur- 
face joining points whose declinations are equal at any given 
time. The isogonic line joining points having no declination 
is called the agonic line. There is such a line crossing the 
United States passing just east of Charleston, S. C, and just 
west of Detroit, Mich. All points east of this line have a 
western declination, and all points west of it have an eastern 
declination. The isogonic lines for 1885 for the whole of 
the United States are shown on Plate I.* It will be noted 
that where the observations are most thickly distributed, 
as in Missouri for instance, there the isogonic lines are most 
crooked ; showing that if the declinations were accurately 
known for all points of this map the isogonic lines would be 
much more irregular, and would be changed very much in 
position in many places. 

The isogonic lines given on this chart are all moving west- 
ward, so that all western declinations are increasing and all 
eastern declinations are decreasing. They are not all moving at 
the same rate, however, those in New Brunswick and also those 
near the eastern boundaries of California and Oregon being 
about stationary. For many points in the United States and 
Canada the rate of change in the declination has been observed, 
and formulae determined for computing the declination for each 
point, which formulae will probably remain good for the next 
twenty years. The following tables t give this information. In 
these tables / is the time in calendar years. Thus for July I, 
1885, /= 1885.5. I n tne fi rst table all the formulae have been re- 



* Reduced from charts in the U. S. Coast and Geodetic Survey Report for 1882, 
f Taken from the above report. 



24 SURVEYING. 



f erred to one date — Jan. I, 1850. Here m is used to represent 
the time in years after 1850, or *#— t — 1850. Thus, for July 1, 
1885, m = 35.5. The annual value of this secular change in the 
declination is marked at various points on the isogonic chart 
given in Plate I., but from the small number of the observa- 
tions, both in time and space, it is evident that no great reli- 
ance can be placed on any such chart for exact information. 

It will be seen that the change in the declination over the 
Northern States will average about one minute to the mile in an 
east and west direction. A value of the declination found in 
one end of a county may be some forty minutes in error in the 
other end of the same county. This shows that the declina- 
tion must be known for the exact locality of the survey. In 
fact, the surveyor can never be sure of his declination until he 
has observed it for himself for the given time and place. This 
is best done by means of a transit instrument, and such a 
method is given in the chapter on Geodetic Surveying. If, 
however, no transit is at hand, a result sufficiently accurate for 
compass surveying may be obtained by the compass itself. 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 2$ 



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SURVEYING. 



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ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 2g 

32. Other Variations of the Declination. — In addition 
to the daily and secular changes in the declination, there are 
others worthy of mention. 

The annual variation is very small, being only about a half- 
minute of arc from the mean position for the year. It may 
therefore be neglected. 

The lunar inequalities are still smaller, being only about fif- 
teen seconds of arc from the mean position. 

Magnetic disturbances are due to what are called magnetic 
storms. They may occur at any time, and cannot be predicted. 
They may last a few hours, or even several days. " The fol- 
lowing table of the observed disturbances, in a bi-hourly series, 
at Philadelphia, in the years 1840 to 1845, w ^ gi ve an idea of 
their relative frequency and magnitude : 



Deviations from nor- 
mal direction. 


Number of 
disturbances. 


3'. 6 to 10'. 8 


2189 


10'. 8 to 18'. 1 


147 


18'. 1 to 25'. 3 


18 


25'. 3 to 32'. 6 


3 




O 



"At Madison, Wis., where the horizontal magnetic intensity 
is considerably less, very much larger deflections have been 
noticed. Thus, on October 12, 1877, one of 48', and on May 
28, 1877, one of i° 24', were observed." * 

The geometric axis of a needle may not coincide with its 
magnetic axis, and hence the readings of two instruments at 
the same station may differ slightly when both are in adjust- 
ment. In this case the declination should be found for each 
instrument independently. 

33. To Find the Declination of the Needle. — The 

- From Report of the U. S. Coast and Geodetic Survey for 1882. 



30 SURVEYING. 



method here given is by means of the compass and a plumb- 
line, and is sufficiently accurate for compass-work. The com- 
pass-sights are brought into line with the plumb-line and the 
pole-star (Polaris), when this is at either eastern or western 
elongation. This star appears to revolve in an orbit of i° 18' 
radius. Its upper and lower positions are called its upper and 
lower culminations, and its extreme east and west positions are 
called its eastern and western elongations, respectively. When 
it is at elongation it ceases to have a lateral component of 
motion, and moves vertically upward at eastern and downward 
at western elongation. If the star be observed at elongation, 
therefore, the observer's watch may be as much as ten or 
fifteen minutes in error, without its making any appreciable 
error in the result. The method of making the observation is 
as follows : 

Suspend a fine plumb-line, such as an ordinary fishing-line, 
by a heavy weight swinging freely in a vessel of water. The 
line should be suspended from a rigid point some fifteen or 
twenty feet from the ground. Care must be taken to see that 
the line does not stretch so as to allow the weight to touch the 
bottom of the vessel. Just south of this line set two stakes in 
the ground in an east and west direction, leaving their tops at 
an elevation of four or five feet. Nail to these stakes a board 
on which the compass is to rest. The top of this board should 
be smooth and level. This compass-support should be as far 
south of the plumb-line as possible, to enable the pole-star to 
be seen below the line-support. A sort of wooden box may 
be provided, in which the compass is rigidly fitted and levelled. 
Several hundred feet of nearly level ground should be open to 
the northward, for setting the azimuth-stake. Prepare two 
stakes, tacks, and lanterns. Find from the table given on page 
32 the time of elongation of the star. About twenty minutes 
before this time, set the compass upon the board, bringing both 
sights in the plane defined by the plumb-line and star. The 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 3 1 

line must be illuminated. The star will be found to move 
slowly east or west, according as it is approaching its eastern 
or western elongation. When it ceases to move laterally, the 
compass is carefully levelled, the rear compass-sight brought 
into the plane of the line and star, and then the forward com- 
pass-sight made to coincide with the rear sight and plumb-line. 
(If the forward sight were tall enough, we could at once bring 
both slits into coincidence with line and star.) Continue to ex- 
amine rear sight, line, and star, and rear sight, forward sight, 
and line alternately, until all are found to be in perfect coinci- 
dence, the instrument still being level. If this is completed 
within fifteen minutes of the true local time of elongation, the 
observation may be considered good ; and if it is completed 
within thirty minutes of the time of elongation, the resulting 
error in azimuth will be less than one minute of arc. Having 
completed these observations, remove the plumb-line and set a 
stake in the line of sight as given by the compass, several hun- 
dred feet away. In the top of this stake a tack is to be set 
exactly on line. For setting this tack, a board may be used, 
having a vertical slit about \ inch wide, covered with white 
cloth or paper, behind which a lamp is held. This slit can 
then be accurately aligned and the tack set. A small stake 
with tack is now set just under the compass (or plumb-line), 
and the work is complete for the night. Great care must be 
taken not to disturb the compass after its final setting on the line 
and star. 

At about ten o'clock on the following day, mount the com- 
pass over the south stake. From the north stake lay off a line 
at right angles to the line joining the two stakes (by compass, 
optical square, or otherwise) towards the west if eastern 
elongation, or towards the east if western elongation had been 
observed. Carefully measure the distance between the two 
stakes by some standardized unit. From the table of azimuths 
on page 33 find the azimuth of the star at elongation for the 



32 



SURVEYING. 



given time and latitude. Multiply the tangent of this angle 
by the measured distances between the stakes, and care- 
fully lay it off from the north tack, setting a stake and tack. 
This is now in the meridian with the south point. With the 
compass in good adjustment, especially as to the bubbles and 
the verticality of the sights, the observation for declination 
may now be made. If this be done at about 10.30 A.M., it 
will give the mean daily declination. Many readings should 
be made, allowing the needle to settle independently each time. 
The fractional part of a division on the graduated limb should 
be read by the declination-vernier, thus enabling the needle to 
be set exactly at a graduation-mark. If all parts of this work 
be well done, it will give the declination as accurately as the 
flag can be set by means of the open sights. 



MEAN LOCAL TIME (ASTRONOMICAL, COUNTING FROM NOON) 
OF THE ELONGATIONS OF POLARIS. 

[The table answers directly for the year 1885, and for latitude -f- 40 .] 



Date. 


Eastern 
Elongation. 


Western 
Elongation. 


Dat 


e. 


Eastern 
Elongation. 


Western 
Elongation. 


Jan. 


1 


Q h 35m 3 


I2 h 24 m .6 


1 J^y 


I 


I2 U 39 m .6 


O h 32 m .8 


n 


15 


23 36 .1 


II 29 .3 


: << 


15 


II 44 .7 


23 34 -o 


Feb. 


1 


22 29 .O 


10 22 .2 


Aug. 


I 


IO 38 .2 


22 27 .5 


1 < 


15 


21 33 -7 


9 27 .0 


CI 


15 


9 43 -3 


21 32 .6 


Mar. 


1 


20 38 .5 


8 31 -8 


Sept. 


I 


8 36 .7 


20 26 .0 


a 


15 


19 43 -4 


7 36 .6 


tt 


15 


7 41 -7 


19 31 .1 


Apr. 


1 


18 36 -4 


6 29 .7 


Oct. 


I 


6 38 .9 


18 28 .2 


<< 


15 


17 41 .4 


5 34 -7 


< c 


15 


5 43 -9 


17 33 -2 


May 


1 


16 38 .6 


4 31 -8 


Nov. 


I 


4 37 .0 


16 26 .4 


(< 


15 


15 43 -7 


3 36 .9 


<< 


15 


3 4i -9 


15 31 -3 


June 


1 


14 37 -I 


2 30 .3 


Dec. 


I 


2 38 .9 


14 28 .2 


<< 


15 


13 42 .2 


1 35 -4 


«{ 


15 


1 43 .6 


13 33 -o 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 33 

AZIMUTH (FROM THE NORTH) OF POLARIS, WHEN AT ELONGA- 
TION, BETWEEN THE YEARS 1886 AND 1895, FOR DIFFERENT 
LATITUDES BETWEEN + 25 AND + 50 . 



Lat. 


1886.0 


1887.0 


i838.o 


1889.0 


1890.0 
/ 


1891.0 


1892.0 


1893.0 


1894.0 


1895.0 





/ 


c / 


/ 


/ 


/ 


t 


/ 


/ 


/ 


+ 25 


1 26.0 


I 25.7 


I 25.3 


1 25.0 


1 24.6 


1 24.3 


1 23.9 


1 23.6 


1 23.2 


1 22.9 


26 


26.7 


26.4 


26.O 


25.7 


25-3 


25.0 


24.6 


24-3 


23-9 


23.6 


27 


27-5 


27.1 


26.8 


26.4 


26.0 


25.7 


25-4 


25.1 


24.7 


24.3 


28 


28.3 


27.9 


27.6 


27.2 


26.8 


26.5 


26.2 


25.8 


25.4 


25-1 


29 


29.1 


28.8 


28.4 


28.0 


27.6 


27-3 


27.0 


26.6 


26.3 


25-9 


30 


30.0 


29.6 


29-3 


28.9 


28.5 


28.2 


27.8 


27-5 


27.1 


26.8 


31 


30.9 


30.5 


30.2 


29.8 


29.4 


29.1 


28.8 


28.4 


28.0 


27.6 


32 


3i-9 


31-5 


31.2 


30.8 


30.4 


30.1 


29.7 


29-3 


29.0 


28.6 


33 


33-o 


32.6 


32.2 


31.8 


3i.4 


3i-i 


30.7 


30.3 


30.0 


29.6 


34 


34-0 


33-6 


33-3 


32.9 


32.5 


32.1 


31.8 


3i-4 


31.0 


30.6 


35 


35-2 


34 8 


34-4 


34-0 


33-6 


33-2 


32.9 


32.5 


32.1 


31.7 


36 


36.4 


36.0 


35-6 


35-2 


34-8 


34-4 


34.o 


33-6 


33-2 


32.9 


37 


37-6 


37-2 


36.8 


36-4 


36.0 


35-6 


35-2 


34-8 


34-5 


34.1 


38 


38.9 


38.5 


38.1 


37-7 


37-3 


36.9 


36.5 


36.1 


35-7 


35-3 


39 


40.3 


39-9 


39-5 


39-i 


38.7 


38.3 


37-9 


37-5 


37-1 


36.7 


40 


41.8 


41.4 


4.1.0 


40.5 


40.1 


39-7 


39-3 


38.9 


38.5 


38.1 


4i 


43.3 


42.9 


42.5 


42.0 


41.6 


41.2 


40.8 


40.4 


40.0 


39- 6 


42 


44.9 


44-5 


44.1 


43-6 


43-2 


42.8 


42.4 


42.0 


4i.5 


41. 1 


43 


46.6 


46.1 


45-7 


45-3 


44-9 


44.4 


44.0 


43-6 


43-2 


42.7 


44 


48.4 


47-9 


47-5 


47.1 


46.6 


46.2 


45-8 


45.3 


44.9 


44-4 


45 


50.3 


49.8 


49.4 


48.9 


48.5 


48.1 


47.6 


47.1 


46. 1 


46.2 


46 


52.2 


51.8 


51.3 


50.9 


50.4 


50.0 


49-5 


49.0 


48.6 


48.2 


47 


54-3 


53-8 


53-4 


52.9 


52.5 


52.0 


51.5 


5i-o 


50.6 


50.2 


48 


56.5 


56.0 


55.6 


55-1 


54-6 


54-2 


53-7 


53-2 


52.8 


52.3 


49 


1 58.8 


1 58.3 


1 57-9 


57-4 


56.9 


56.5 


56.0 


55-5 


55-0 


54-5 


+ 50 


2 01.3 


2 00.8 


2 00.3 


1 59-8 


1 59-3 


1 58.8 


1 58.4 


1 57-9 


1 57-4 


1 56.9 



34 SURVEYING. 



If the elongation of Polaris does not come at a suitable 
time for observing for declination, the upper culmination, which 
occurs 5 h 54 m .6 after the eastern, or the lower culmination, 
6 h 03 m .4 after the western elongation, may be chosen. The 
objection to this is that the star is then moving at its most 
rapid rate in azimuth. It is so near the pole, however, that if 
the observation can be obtained within two minutes of the 
time of its culmination the resulting error will be less than i' 
of arc. This will then give the true meridian without having 
to make offsets. 

It must be remembered that the time of elongation given 
in the table is the local time at the place of observation. In- 
asmuch as hourly meridian time is now carried at most points 
in this country to the complete exclusion of local time, it will 
be necessary to find the local time from the known meridian or 
watch time. Thus, all points in the United States east of Pitts- 
burgh use the fifth-hour meridian time (75 w. of Greenwich) ; 
from Pittsburgh to Denver, the sixth-hour meridian time (90 
w. of Greenwich), etc. To find local time, therefore, the longi- 
tude east or west of the given meridian must be found. This 
can be determined with sufficient accuracy from a map, Thus, 
if the longitude of the place is 8o° w. from Greenwich, it is 
5° w. of the fifth-hour meridian, or local time is twenty min- 
utes slower than meridian time at that place If meridian time 
is used at such a place, the elongation will occur twenty min- 
utes later than given by the table. If the longitude from 
Washington is given on the map consulted, add it to 77 if 
west of Washington, and subtract it from JJ° if east of Wash- 
ington, to get longitude from Greenwich. 

USE OF THE NEEDLE-COMPASS. 

34. The Use of the Needle-compass is confined almost 
exclusively to land-surveying, where an error of one in three 
hundred could be allowed. As the land enhances in value, 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 35 

however, there is an increasing demand for more accurate 
means of determining areas than the compass and chain af- 
ford. The original U. S. land-surveys were all made with the 
needle, or with the solar, compass and Gunter's chain. Hence 
all land boundaries in this country have their directions given 
in compass-bearings, and their lengths in chains of sixty-six 
feet each. 

The compass is used, therefore, — 

1. To establish a line of a given bearing. 

2. To determine the bearing of an established line. 

3. To retrace old lines. 

If the true bearing is to be used, the declination of the 
needle from the meridian must be determined and set off by 
the vernier. 

If the magnetic bearing is used, the declination of the 
needle at the time the survey was made should be recorded 
on the plat. 

If old lines are to be retraced, the declinations at the times 
of both surveys must be known. 

The needle should be read to the nearest five minutes. 
This requires reading to sixths of the half-degree spaces, but 
this can be done with a little practice. 

Always lift the needle from the pivot before moving the in- 
strument. 

If the needle is sluggish in its movements and settles quickly 
it has either lost its magnetic force or it has a blunt pivot. In 
either case it is likely to settle considerably out of its true posi- 
tion. The longer a needle is in settling the more accurate will 
be its final position. It can be quickly brought very near its 
true position by checking its motion by means of the lifting 
screw. In its final settlement, however, it must be left free. 

Careful attention to the instrumental adjustments, to local 
disturbances, and close reading of the needle are all essential 
to good results with the compass. 



36 SUR VE YING. 



35. To set off the Declination, we have only to remem- 
ber that the declination arc is attached to the line of sight and 
that the vernier is attached to the graduated circle. If the 
declination is west, then when the line of sight is north the 
north end of the needle points to the left of the zero of the 
graduated circle. In order that it may read zero, or north, the 
circle must be moved towards the left, or opposite to the hands 
of a watch. On the other hand, if the declination is east, the 
circle to which the^vernier is attached should be moved with 
the hands of a watch. This at once enables the observer to 
set the vernier so that the needle-readings will be the true 
bearings of the line of sight. 

36. Local Attractions may disturb the needle by large or 
small amounts, and these often come from unknown causes. 
The observer should have them constantly in mind, and keep all 
iron bodies at a distance from the instrument when the needle 
is being read. The glass cover may become electrified from 
friction, and attract the needle. This can be discharged by 
touching it with a wet finger, or by breathing upon it. Read- 
ing-glasses should not have gutta-percha frames, as these be- 
come highly electrified by wiping the lens, and will attract the 
needle. Such glasses should have brass or German-silver 
frames. No nickel coverings or ornaments should be near, as 
this metal has magnetic properties. A steel band in a hat- 
brim, or buttons containing iron, have been known to cause 
great disturbance. In cities and towns it is practically impos- 
sible to get away from the influence of some local attraction, 
such as iron or gas pipes in the ground, iron lamp-posts, fences, 
building-fronts, etc. For this reason the needle should never 
be used in such places. 

In many regions, also, there are large magnetic iron-ore de- 
posits in the ground, which give special values for the declina- 
tion at each consecutive station occupied. It is practically 
impossible to use magnetic bearings in such localities. 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 3/ 

The test for local attraction in the field-work is to read the 
bearing of every line from both ends of it. If these are not 
the same, and no error has been made, there is some local dis- 
turbance at one station not found at the other. If there is 
known to be mineral deposits in the region it may perhaps be 
laid to that. If not, the preceding station should be occupied 
again, and the cause of the discrepancy inquired into. If the 
forward and reverse bearings of all lines agree except the bear- 
ings taken from a single station, then it may be assumed there 
is local attraction at that station. 

37. To establish a Line of a Given Bearing, set the com- 
pass up at a point on the line, turn off the declination on the 
declination-arc, and bring the north end of the needle to the 
given bearing. The line of sight now coincides with the re- 
quired line, and other points can be set. 

38. To find the True Bearing of a Line, set the compass 
up on the line, turn off the declination by the vernier, bring 
the line of sight to coincide with the line with the south part 
of the graduated circle towards the observer, and read the 
north end of the needle. This gives the forward bearing of 
the line. 

39. To retrace an Old Line, set the compass over one 
well-determined point in the line and turn the line of sight 
upon another such point. Read the north end of the needle. 
If this reading is not the bearing as given for the line, move 
the vernier until the north end of the needle comes to the 
given bearing, when the sights are on line. The reading of 
the declination-arc will now give the declination to be used in 
retracing all the other lines of the same survey. If a second 
well-determined point cannot be seen from the instrument-sta- 
tion, a trial-line will have to be run on an assumed value for 
the declination, and then the value of the declination used on 
the first survey computed. Thus, if the trial-line, of length /, 
comes out a distance x to the right of the known point on 



38 



SUR VE YING. 



the line, the vernier is to be moved in the direction of the hands 

x 
of a watch an angular amount whose tangent is j. If the 

trial-line comes out to the left of the point, move the vernier 
in a direction opposite to the hands of a watch. 

PRISMATIC COMPASS. 

40. The Prismatic Compass is a hand-instrument pro- 
vided with a glass prism so adjusted that the needle can be 
read while taking the sight. A convenient form is shown in 
Fig. 9, which is carried in the pocket as a watch. The line of 




Fig. 9. 



sight is established by means of the etched line on the glass 
cover S. It is used in preliminary and reconnoissance work, in 



clearing out lines, etc. 



EXERCISES FOR COMPASS ALONE OR FOR COMPASS AND CHAIN. 

41. Run out a line of about a mile in length, on somewhat uneven ground, 
establishing several stations upon it, using a constant compass-bearing. Then 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 39 

run back by the reverse bearings and note how nearly the points coincide with the 
former ones. The chain need not be used. 

42. Select some half dozen points that enclose an area of about forty acres 
(one quarter mile square) on uneven ground. Let one party make a compass- 
and-chain survey of it, obtaining bearing and length of each side. Then let 
other parties take these field-notes and, all starting from a common point, run 
out the lines as given by the field-notes, setting other stakes at all the remaining 
corners, each party leaving special marks on their own stakes. Let each party 
plot their own survey and compare errors of closure. 

43. Select five points, three of which are free from local attraction, while two 
consecutive ones are known to be subject to such disturbance. Make the sur- 
vey, finding length and forward and reverse bearings of every side. Determine 
what the true bearing of each course is, and plot to obtain the error of closure. 

44. Let a number of parties observe for the declination of the needle, using 
a common point of support for the plumb-line. Let each party set an inde- 
pendent meridian stake in line with the common point. Note the distance of 
each stake from the mean position, and compute the corresponding angular dis- 
crepancies. (March and September are favorable months for making these 
observations, for then Polaris comes to elongation in the early evening.) 

The above problems are intended to impress upon the student the relative 
errors to which his work is subject. 

THE SOLAR COMPASS. 

45. The Burt Solar Compass essentially consists first, of 
a polar axis rigidly attached in the same vertical plane with a 
terrestrial line of sight, the whole turning about a vertical axis. 
When this plane coincides with the meridian plane, the polar 
axis is parallel with the axis of the earth. Second, attached 
to the polar axis, and revolving about it, is a line of collimation 
making an angle with the polar axis equal to the angular dis- 
tance of the sun for the given day and hour from the pole. 
This latter angle is 90 plus or minus the sun's declination, 
according as the sun is south or north of the equator. The 
polar axis must therefore make an angle with the horizon 
equal to the latitude of the place, and the line of collimation 
must deviate from a perpendicular to this axis by an angular 
amount equal to, and in the direction of, the sun's declination. 
With these angles properly set, and the line of collimation 



40 



SURVEYING. 



turned upon the sun, the vertical plane through the terrestrial 
line of sight, and the polar axis must lie in the meridian, for 




Fig 



otherwise any motion of the line of collimation about its axis 
would not bring it upon the sun. 

In Fig. 10 is shown a cut of this instrument as manufac- 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 4 1 

tured by Young & Sons, Philadelphia. The polar axis is shown 
at/, and the terrestrial line of sight is defined by the slits in 
the vertical sights, the same as in the needle-compass. The 
line of collimation is defined by a lens at the upper end of the 
arm a, and a silver plate at the lower end, containing gradua- 
tions with which the image of the sun, as formed by the lens, 
is made to coincide. The polar axis is given the proper incli- 
nation by means of the latitude-arc /, and the line of collima- 
tion is inclined from a perpendicular to this axis by an amount 
equal to the sun's declination by means of the declination-arc 
d. When these arcs are properly set, the arm a is revolved 
about the polar axis, and the whole instrument about its verti- 
cal axis, until the image of the sun is properly fixed on the 
lines of the silver plate, when the terrestrial line of sight, as 
defined by the vertical slits, lies in the true meridian. Any 
desired bearing may now be turned off by means of the hori- 
zontal circle and vernier, shown at v. The accuracy with 
which the meridian is obtained with this instrument depends 
on the time of day, and on the. accuracy with which the lati- 
tude- and declination-angles are set off. It is necessary to at- 
tend carefully, therefore, to the 



ADJUSTMENTS OF THE SOLAR COMPASS. 

46. To make the Plane of the Bubbles perpendicular to 
the Vertical Axis. — This is done by reversals about the verti- 
cal axis, the same as with the needle-compass. 

47. To adjust the Lines of Collimation.— The declination- 
arm a has two lines of collimation that should be made paral- 
lel. As it is shown in the figure, it is set for a south declina- 
tion. This is the position it will occupy from Sept. 20 to 
March 20. When the sun has a north declination, as from 
March 20 to Sept. 20, the declination-arm is revolved 180 
about the polar axis, and a line of collimation established by 



42 SUR VE YING. 



a lens and a graduated disk on opposite ends from those pre- 
viously used. Each end of this arm, therefore, has both a lens 
and a disk, each set of which establishes a line of collimation. 
The second adjustment consists in making these two lines, of col- 
limation parallel to each other. They are made parallel to each 
other by making both parallel to the faces of the blocks con- 
taining the lenses and disks. To effect this, the arm must be 
detached and laid upon an auxiliary frame which is attached 
in the place of the arm, and which is called an adjuster. With 
the latitude- and declination-arc set approximately for the given 
time and place, lay the declination-arm upon the adjuster, and 
bring the sun's image upon the disk. Now turn the arm care- 
fully bottom side up (not end for end) and see if the sun's 
image comes between the equatorial lines on the disk.* If not, 
adjust the disk for one half the displacement, and reverse again 
for a check. When this disk is adjusted, turn the arm end for 
end, and adjust the other disk in a similar manner. Having 
now made both lines of collimation parallel to the edges of the 
blocks, they are parallel to each other. 

48. To make the Declination-arc read Zero when the 
Line of Collimation is at Right Angles to the Polar Axis. 
— Set the vernier on the declination-arc to read zero. By any 
means bring the line of collimation upon the sun. When 
carefully centred on the disk, revolve the arm 180 quickly 
about the polar axis, and see if the image now falls exactly on 
the other disk. If not, move the declination-arm by means of 
the tangent-screw until the image falls exactly on the disk in 
both positions. Read the declination-arc, loosen the screws in 
the vernier-plate, and move it back over one half its distance 
from the zero-reading. Centre the image again, reverse 180 , 
and test. This adjustment depends on the parallelism of the 
two lines of collimation. If the vernier-scale is not adjustable, 

* It would not be expected to fall between the hour-lines on the disk, since 
some time has elapsed. 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 43 

one half the total movement is the index error of the declina- 
tion-arc, and must be taken into account in all settings on this 
arc. 

The two preceding adjustments should be made near the 
middle of the day. 

49. To adjust the Vernier of the Latitude-arc. — Find the 
latitude of the place, either from a good map or by a transit- 
observation. Set up the compass a few minutes before noon, 
with the true declination se't off for the given day and hour. 
Bring the line of collimation upon the sun, having it clamped 
in the plane of the sights, or at the twelve-hour angle, and 
follow it by moving the latitude-arc by means of the tangent- 
screw, and by turning the instrument on its vertical axis. 
When the sun has attained its highest altitude, read the lati- 
tude-arc. Compare this with the known latitude. Move the 
vernier on this arc until it reads the true latitude ; or, if this 
cannot be done, the difference is the index error of the latitude- 
arc. If, however, the latitude used with the instrument be 
that obtained by it, as above described, then no attention need 
be paid to this error. This error is only important when the 
true latitude is used with the instrument in finding the meridian, 
or where the true latitude of the place is to be found by the in- 
strument. In using the solar compass, therefore, always use 
the latitude as given by that instrument by a meridian observa- 
tion on the sun* 

50. To make the Terrestrial Line of Sight and the Polar 
Axis lie in the same Vertical Plane. — This should be done by 
the maker. The vertical plane that is really brought into the 
meridian by a solar observation is that containing the polar 
axis, and by as much as the plane of the sights deviates from 

* Since the sun may cross the meridian as much as 15 minutes or more 
before or after mean noon, this observation may have to be taken that much 
before or after 12 o'clock mean time. It is, however, in all cases, an observation 
on the sun ai culmination. 



44 SURVEYING. 



this plane, by so much will all bearings be in error. The best 
test of this adjustment is to establish a true meridian by the 
transit by observations on a circumpolar star ; and then by 
making many observations on this line, in both forenoon and 
afternoon, one may determine whether or not the horizontal 
bearings should have an index-correction applied. 

USE OF THE SOLAR COMPASS. 

51. The Solar Compass is used on land and other surveys 
where the needle-compass is either too inaccurate, or where, 
from local attraction, the declination of the needle is too vari- 
able to be accurately determined for all points in the survey. 
Where there is no local attraction, however, and the declination 
of the needle is well known, the advantages of the solar com- 
pass in accuracy are fairly offset by several disadvantages in its 
use which do not obtain with the needle-compass. Thus, the 
solar compass should never be used when the sun is less than 
one hour above the horizon, or less than one hour from noon. 
Of course it cannot be used in cloudy weather. For such times 
as these bearings may be obtained by a needle which is always 
attached, but then the instrument becomes a needle-com- 
pass simply. It is also much more trouble, and consumes 
more time in the field than the needle-compass. But more 
significant than any of these is the fact that if the adjustments 
are not carefully attended to, the error in the bearing of a line 
may be much greater by the solar compass than is likely to 
be made by the needle-compass, when there is no local attrac- 
tion. It is possible, however, to do much better work with 
the solar compass than can be done with the needle-com- 
pass. 

52. To find the Declination of the Sun. — On account of 
the inclination of the earth's axis to the plane of its orbit, the 
sun is seen north of the celestial equator in summer, and south 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 45 



of it in winter. This deviation, north or south of the equator, 
is called north or south declination, and is measured from any 
point on the earth's surface in degrees of arc. 

On about the 2ist of June the sun reaches its most northern 
declination, and begins slowly to return. Its most southern 
point is reached about December 2 1st. In June and Decem- 
ber, therefore, the sun is changing its declination most slowly, 
while at the intervening quadrant-points of the earth's orbit, 
March and September, it is changing its declination most 
rapidly, being as much as one minute in arc for one hour in 
time. It is evident, therefore, that we must regard the decli- 
nation of the sun as a constantly changing quantity, and, 
for any given day's work, a table of declinations must be 
made out for each hour of the day. The American Ephemeris 
and Nautical Almanac gives the declination of the sun for noon 
of each day of the year for both Greenwich and Washington. 
Since the time universally used in this country is so many 
hours from Greenwich, it is best to use the Greenwich declina- 
tions. Since, also, we are five, six, seven, or eight hours west 
of Greenwich, the declination given in the almanac for Green- 
wich noon of any day will correspond to the declination here 
a t 7 j 6, 5, or 4 o'clock A.M. of the same date, according as East- 
ern, Central, Mountain, or Western time is used. As this 
standard time is seldom more than 30 minutes different from 
local time, and as this could never affect the declination by more 
than 30 seconds of arc, it will here be considered sufficient £0 
correct the Greenwich declination by the change, as found for 
the standard time used. Thus, if Central (90th meridian) time 
is used, the declination given in the almanac is the declination 
at 6 o'clock A.M. at the place of observation. To this must be 
added (algebraically) the hourly change in declination, which is 
also given in the almanac. A table may thus be prepared, giv- 
ing the declination for each hour of the day. 

53. To correct the Declination for Refraction. — All rays 



46 SURVEYING. 



of light coming to the earth from exterior bodies are refracted 
downward, thus causing such bodies to appear higher than 
they really are. This refraction is zero for normal (vertical) 
lines, and increases towards the horizon. It varies largely, 
also, with the special temperature, pressure, and hydrometrical 
condition of the atmosphere. Tables of refraction give only 
the mean values, and these may differ largely from the values 
found for any given time, especially for lines near the horizon. 
It is for this reason that all astronomical observations made 
near the horizon are very uncertain. There is but one setting 
on the solar compass that has reference to the position of the 
sun in the heavens, and that is the declination. Now, the re- 
fraction changes the apparent altitude of the body ; and by so 
much as a change in the altitude changes the declination, by 
so much does the apparent declination differ from the true dec- 
lination. Evidently it is the apparent declination that should 
be set off. When the sun is on the meridian, the change in 
altitude has its full effect in changing the declination, but at 
other times the change in declination is less than the change 
in altitude. 

The correction to the declination due to refraction is 



* 



C = 57" cot (S + N), 

where & = declination, plus when north and minus when south. 
iVis an auxiliary angle such that 

tan N = cot cos t, 

where is the latitude, and t is the hour-angle from the meridian. 
The following table is computed by this formula : 

* See Chauvenet's "Spherical and Practical Astronomy," vol. i. p. 171. 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 47 



TABLE OF CORRECTIONS TO BE APPLIED TO THE DECLINA- 
TION FOR REFRACTION.* 





DECLINATIONS. 


Hour 
Angle. 


For Latitude 30 




+ 20 


+ 15° 


+ IO° 


+ 5° 


o° 


-5° 


— IO° 


-15° 


— 20°- 


oh. 


10" 


15" 


21" 


27" 


33" 


40" 


48" 


57" 


i'o8" 


2 


14 


19 


25 


3 1 


38 


46 


54 


1 '05 


1 18 


3 


20 . 


26 


32 


39 


47 


55 


i'o6 


1 19 


1 36 


4 


32 


39 


46 


52 


i'o6 


l'lQ 


1 35 


1 57 


2 29 


5 


x'oo 


i'io 


l'24 


l'52 


2 07 


244 


3 46 


5 43 


13 06 



For Latitude 32 30'. 



oh. 


13" 


18" 


24" 


30" 


36" 


44" 


52" 


I '02" 


l'l 4 " 


2 


17 


22 


28 


35 


42 


50 


I '00 


I II 


I 26 


3 


23 


29 


35 • 


43 


5i 


i'oi 


1 J 3 


x 28 


1 47 


4 


35 


, 43 


5i 


i'oi 


i'i3 


I 27 


I 46 


1 13 


2 54 


5 


i'o3 


i'i5 


i'3i 


153 


2 20 


305 


425 


736 





For Latitude 35= 



oh. 


15" 


21" 


27" 


33" 


40" 


48" 


57" 


i'o8" 


l'2l" 


2 


20 


25 


32 


38 


46 


55 


1 '05 


1 18 


1 35 


3 


26 


33 


39 


47 


56 


1 '07 


1 21 


138 


2 00 


4 


39 


47 


56 


1 '07 


l'20 


1 36 


1 59 


2 32 


325 


5 


x'07 


l'20 


i's8 


2 OO 


2 ; 34 


3 29 


5 14 


10 16 





For Latitude 37 30'. 



oh. 


18" 


24" 


30" 


36" 


44" 


52" 


1 '02" 


i'i 4 " 


l'2Q" 


2 


22 


28 


35 


42 


50 


I 'OO 


1 12 


1 26 


1 45 


3 


29 


36 


43" 


52 


1 '02 


I 14 


1 29 


1 49 


2 16 


4 


, 43 


5i 


i'oi 


i'i3 


T 27 


I 49 


2 14 


2 54 


4 05 


5 


i'ii 


l'26 


1 54 


2 10 


249 


3 55 


615 


1458 .. 





For Latitude 40 . 



oh. 


21" 


27" 


33" 


40" 


48" 


57" 


i'o8" 


l'2l" 


i'39" 


2 


25 


32 


39 


46 


, 52 


i'o6 


1 19 


1 35 


* 57 


3 


33 


40 


48 


57 


i'o8 


1 21 


138 


2 02 


2 3 6 


4 


, 47 


55 


i'o6 


x'lg 


1 36 


158 


2 30 


3 21 


4 59 


5 


i'i5 


i'3i 


1 5i 


2 20 


3°5 


425 


7 34 


25 18 





For Latitude 42 30' 



oh. 


24" 


30" 


36" 


44" 


52" 


iW 


i'i 4 " 


l'2 9 " 


i' 49 " 


2 


28 


35 


39 


50 


1 '00 


I 12 


1 26 


1 45 


2 11 


3 


36 


43 


52 


1 '02 


1 13 


1 29 


1 49 


2 17 


2 59 


4 


50 


1 '00 


i'ii 


1 26 


1 44 


2 IO 


2 49 


3 55 


6 16 


5 


i'i6 


1 36 


158 


2 30 


3 22 


5 00 


924 







* This table was computed by Edward W. Arms, C.E., for W. & L. E. Gurle]', Troy, N. Y., 
and is here given by the kind permission of the latter. 



48 



SURVEYING. 











DECLINATIONS. 








Hour 








For Latitude 45*. 








Angle. 


















+ 20° 


+ 15° 


+ IO° 


+ 5* 


o° 


-5° 


- IO° 


-15° 


— 20° 


oh. 


27" 


33" 


40" 


48" 


57" 


i'o8" 


l'ai" 


i' 39 " 


2'02" 


2 


3 2 


39 


46 


52 


i'o6 


1 19 


1 35 


1 57 


2 29 


3 


40 


47 


56 


1 '07 


1 21 


138 


2 00 


2 34 


3 29 


4 


54 


1 '04 


i'i6 


1 33 


1 54 


2 24 


3 " 


4 38 


815 


5 


l'23 


1 41 


2 05 


2 41 


3 40 


5 40 


12 02 







For Latitude 47 30'. 



oh. 


30" 


36" 


44" 


52" 


iW 


i'i 4 " 


i'2 9 " 


i'49" 


2 'l8" 


2 


35 


42 


50 


i'oo 


I 12 


1 26 


1 45 


2 01 


2 51 


3 


43 


5i 


i'oi 


1 13 


I 28 


1 47 


2 15 


2 56 


408 


4 


56 


1 '09 


I 23 


I 40 


205 


2 40 


3 39 


5 37 


11 18 


5 


l'27 


1 46 


2 12 


2 52 


4 01 


6 30 


16 19 







For Latitude 50 



oh. 


33" 


40" 


48" 


57" 


i'o8" 


l'2l" 


i'39" 


2'02" 


2' 3 6" 


2 


38 


46 


55 


i'o6 


1 18 


1 35 


1 57 


2 28 


3 19 


3 


47 


56 


1 '06 


1 19 


1 36 


2 29 


231 


3 23 


5 02 


4 


l'o2 


i'i 4 


1 29 


1 48 


2 16 


258 


4 18 


659 


19 47 


5 


I 30 


1 5i 


2 19 


304 


4 22 


7 28 


24 10 







For Latitude 52 30' 



oh. 


36" 


44" 


52" 


l'02" 


i'i 4 " 


1 '29" 


i' 49 " 


2'l8" 


3'° 5 " 


2 


43 


50 


z 59 


I II 


1 26 


1 42 


2 23 


2 49 


3 55 


3 


50 


i'oo 


I'll 


I 26 


1 45 


2 11 


2 51 


258 


6 22 


4 


1 '05 


1 18 


135 


2 IO 


2 28 


3 19 


4 53 


8 42 




5 


134 


156 


2 27 


3 16 


4 47 


852 









For Latitude 55°. 



oh. 


40" 


48" 


57" 


i'o8" 


l'2l" 


i'39" 


2'02" 


2' 3 6" 


3'33" 


2 


46 


55 


1 '05 


1 18 


1 34 


156 


2 30 


3 15 


4 47 


3 


, 55 


1 '06 


1 19 


1 35 


158 


2 30 


3 21 


4 58 


919 


4 


i'io 


1 23 


1 42 


2 06 


2 43 


3 44 


5 49 


12 41 




5 


137 


2 01 


2 34 


328 


5 15 


10 18 









For Latitude 57° 30' 



oh. 


A a" 

44 


52" 


iW 


i'i 4 " 


1 '29" 


i'49« 


2 'l8" 


3'°5" 


4'3?" 


2 


50 


z 59 


I II 


125 


1 43 


2 09 


247 


3 5* 


6 04 


3 


58 


i'io 


I 24 


1 42 


2 07 


243 


3 45 


5 5o 


1247 


4 


i'ii 


I 25 


1 43 


2 10 


2 50 


5 55 


6 14 


1449 




5 


1 41 


2 06 


2 42 


3 42 


5 46 


12 26 









The corrections given in the table are to be added to all 
north declinations and subtracted from all south declinations. 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 49 



54. Errors in Azimuth due to Errors in the Declina- 
tion and Latitude Angles. — The spherical triangle involved 
in an observation by the solar compass is shown 
in Fig. ii, where Pis the pole, Z the zenith, and <cj 
5 the sun. Then 



the angle at 



P=t, the hour-angle from the 
meridian ; 
" " Z = A, the azimuth from the north 

point ; 
" " 5 = q, the variable or parallactic 

angle. 
Also, the arc PZ '= the co-latitude = 90 — <p ; 

" PS = the co-declination = 90 — $ ; 
" ZS = the co-altitude, or zenith dis- ®V 
tance = 90 — A. 




Fig. 



Taking the parenthetical notation of the figure, we have, 
from spherical trigonometry, 

cos (a) = cos (c) cos (3) + sin (c) sin (b) cos (A), 
But in terms of S, (f>, h, and A, this becomes 

sin 6 =. sin sin h — cos <p cos h cos A. . . . (1) 

In a similar manner, from two other fundamental equations 
of the spherical triangle, we may write 



cos & cos t = cos sin h -f- sin <p cos h cos A ; (2) 

cos S sin t = cos -A sin A, (3) 

If we differentiate equation (1) with reference to A and S, 

4 



50 SURVEYING. 



and then with reference to A and 0, we obtain, after some 
reductions by the aid of equations (2) and (3) 

•M t = — ^i— ( 4 ) 

cos sin / yrT/ 

and dA+ = -^ — -, (5) 

cos tan t ' 

Now, if the change (or error) in 3 and be taken as 1 minute 
of arc, or, in other words, if the settings for declination or lati- 
tude be erroneous by that amount, either from errors in the 
instrumental adjustments or otherwise, then equations (4) and 
(5) show what is the error due to this cause in the azimuth, or 
in the direction of the meridian, as found. In the following 
table, values of dA& and dA§ are given for various values of 
and / (latitude and hour-angle). In this table no attention 
is paid to signs, as it is intended mainly to show the size of the 
errors to which the work is liable from inaccurate settings for 
declination and latitude; the values may, however, be used as 
corrections to the observed azimuths from such inaccuracies by 
observing the instructions in the appended note. 

* dA& signifies the change in A due to a small change, dd, in 8, the other 
functions involved in equation (i) remaining constant. Similarly for dA^, 
when (p alone changes. The derivation of equations (4) and (5) involves a 
knowledge of the infinitesimal calculus. 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 5 1 



TABLE OF ERRORS IN AZIMUTH (BY SOLAR COMPASS) FOR i' 
ERROR IN DECLINATION OR LATITUDE. 



Hour. 


For i' Error in Declination. 


For i' Error in Latitude. 


Lat. 30 


Lat. 40* 


Lat. 50 


Lat. 30 


Lat. 40 


Lat. 50 


II.30 A.M. \ 

I2.3O P.M. ) * ' 


8'.85 


io'.o 


I2'.9 


8'.77 


9 '. 9 2 


Il'.8 


II A.M. ) 
I P.M. ) ' 


4.46 


5 05 


6 .01 


4-33 


4.87 


5.80 


10 A.M. ) 
2 P.M. ) 


2 .31 


2 .6l 


3- II 


2 .00 


2 .26 


2 .70 


9 A.M. ) 

3 p.m. ) 


1.63 


I.85 


2 .20 


1 -15 


I -SO 


1.56 


8 A.M. £ 
4 P.M. ) " 


1-34 


I- 51 


I .80 


0.67 


0.75 


O.9O 


7 A.M. ) 

5 p.m. J 


1 .20 


i-35 


1. 6l 


0.31 


0.35 


0.37 


6 A.M. ) 
6 P.M. ) ' 


1. 15 


1 .30 


I.56 


.0.00 


O.OO 


O .OO 



Note. — Azimuths observed with erroneous declination or co-latitude may be 
corrected by this table by observing that for the line of collimation set too high, 
the azimuth of any line from the south point in the direction S.W.N. E. is 
found too small in the forenoon and too large in the afternoon by the tabular 
amounts for each minute of error in the altitude of the solar line of sight. The 
reverse is true for this line set too low. 

Several important conclusions may be drawn from this table 
and from equations (4) and (5). 

First — That the solar compass should never be used between 
11 A.M. and 1 P.M., and preferably not between 10 A.M. and 2 
P.M., if the best results are desired. 

Second — That at 6 o'clock A.M. and P.M., when the line of col- 
limation lies in a plane at right angles to the plane of the me- 
ridian, no small change in the latitude-arc will affect the accu- 
racy of the result. 



52 SURVEYING. 



Third — From equation (4) and (5), we see that both errors 
have the same sign. Therefore, if the declination-angle be er- 
roneously set off, and the latitude-angle be also affected by an 
equal error in the opposite direction, then the two resulting 
errors in azimuth will tend to compensate. From the table it 
may be seen that for the same latitude and hour-angle they 
would nearly balance each other numerically. If, therefore, 
the declination-angle be affected by an error, and the latitude 
of the place then found by a meridian observation with the com- 
pass, the error of the declination will appear in the resulting lati- 
tude, with the opposite sign. In this way any constant error 
in the declination-angle may be nearly eliminated. 

Fourth — The best times of day for using the solar compass are 
from 7 to 10 A.M. and from 2 to 5 P.M. So far as the instru- 
mental errors are concerned, the greater the hour-angle the 
better the observation ; but when the sun is near the horizon, 
the uncertainties in the refraction may cause unknown errors 
of considerable size. 

Fifth — That for a given error in the setting for declination or 
latitude the resulting error in azimuth will have opposite signs 
in forenoon and afternoon. For, in equations (4) and (5), the 
hour-angle, t, has different signs before and after noon ; and 
therefore sin t and tan t change sign, thus changing the sign 
of the expression. If, therefore, a io-o'clock azimuth is in 
error $' in one direction from erroneous settings, a 2-o'clock 
observation with the same instrument should give an azimuth 
5' in error in the opposite direction. 

55. Solar Attachments are appliances fitted upon transit- 
instruments for the purpose of finding the meridian, the same 
as is done by the solar compass. The principles of construc- 
tion and use are the same as those of the solar compass, the 
application of these principles being quite various, however, 
giving rise to several forms of attachments, some of which will 
be discussed in connection with the transit. Their adjust- 
ments and limitations are nearly the same as those here given. 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 53 



EXERCISES WITH THE SOLAR COMPASS. 

56. Determine a true meridian line by an observation on a circumpolar 
star or otherwise, by either the compass or transit. Set the solar compass up 
on one point of this line with a target set at another point on the established 
meridian. Having carefully adjusted the compass, set the declination-arc to the 
right angle for the given day and hour, corrected for refraction, and make a 
meridian observation on the sun for latitude. If the true latitude of the place is 
known, the difference will be the index error of the latitude-arc. Leave the lati- 
tude-arc set at the readings obtained by the meridian observation (whether it is 
the true latitude or not), and make a series of determinations of the meridian by 
the compass at various times of day. These will usually be in error from the 
true meridian by small amounts. Determine the size of these errors by turn- 
ing upon the target and reading the horizontal circle. Record these errors, 
with the time of day and name of observer. Each student should make a 
series of such observations, determining for himself the errors to which the 
work is liable. The same meridian may be used for all, after it has been prop- 
erly checked by duplicate observations. 

57. Set the latitude- or declination-angle say 3' from its true value, and 
observe at various hours of the day, and see if the resulting errors in azimuth 
are about three times those given in the table. Note that these resulting errors 
are in opposite directions and equal in amount in fore- and after-noon observa- 
tions. 

58. With the solar compass on the meridian as before, select a series of 
points, six or more, which are fixed and plainly visible through the slits. Find 
the bearing to each of these points by a separate observation on the sun in 
each case, paying no attention to the target on the true meridian. Remove the 
solar compass and let another student, ignorant of the first bearings, set the 
ordinary needle compass over the same point. Bring the line of sight upon 
the target and make the needle read south by moving the vernier on the decli- 
nation-arc. In other words, set off the declination of the needle. The bear- 
ings given by the needle compass should now agree with those obtained by the 
solar compass. Read upon the series of selected points, obtaining the bear- 
ings to the nearest five minutes. Let a third student take a transit (or the solar 
compass would do) and find the true bearings of the selected points with refer- 
ence to the established meridian. Compare results and so obtain some data 
for determining the relative accuracy of the solar and the needle-compass. 
The mean of two azimuths by the solar compass taken on the same line at 
equal intervals from noon should be the true azimuth of the line if the instru- 
ment has not changed its adjustments in the mean time. This is the way to 
find the trua azimuth of a line by the solar compass. 

59. Run a line over a series of points (six or more) in the forenoon by the 



54 SURVEYING. 



solar compass, and determine the bearings. In the afternoon run it back again, 
using the bearings obtained in the forenoon, set other stakes where the points are 
not coincident with the old ones, and note the residual discrepancy at the close 
of the work. Divide this by twice the length of the line, and this is the error 
of closure due to erroneous bearings. The chain may be used on the first run- 
ning of the line, but on the retracing the stakes may be set opposite the first 
ones, if not coincident. The object is to determine how much of the error of 
closure in surveying may be attributed to erroneous bearings. 

Do the same with the needle-compass and compare results. The points 
need not be in line, nor need they enclose an area. 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 55 



CHAPTER III. 
INSTRUMENTS FOR DETERMINING HORIZONTAL LINES. 

PLUMB-LINE AND BUBBLE. 

60. The Plumb-line and the Bubble-tube are at once the 
most simple, universal, and essential of all appliances used in 
surveying and astronomical work. Without them neither the 
zenith nor the horizon could be effectually determined, and the 
determination of altitudes and of horizontal lines and planes 
would be out of the question. Even azimuths, bearings, and 
horizontal angles require that the circle by which they are ob- 
tained shall be brought into a horizontal position. 

The direction of the plumb-line is by definition a vertical 
line, pointing to the zenith, and a plane at right angles to this 
line is for that point a horizontal plane. As no two plumb- 
lines can be parallel, so no two planes, respectively horizontal 
at two different positions on the earth's surface, can be par- 
allel. 

Parallel horizontal planes can only be planes at different 
elevations, all horizontal for a single position on the earth's 
surface. 

A level surface is a surface (not a plane) which is at every 
point perpendicular to a plumb-line at that point. If the 
earth were covered with a fluid in a quiescent state, the sur- 
face of this fluid would be a level surface. This surface would 
not be a true oblate spheroid, but would in places vary several 
hundred feet from such a mean spheroidal surface. This is 
owing to the fact that the earth is not a homogeneous body, 



5 6 SURVEYING. 



thus causing the centre of mass to deviate from the centre of 
form. Owing also to much irregularity in the distribution of 
the mass, with respect to the form of the earth, there are many- 
irregular deviations of the plumb-line* from any one point. A 
level surface follows all such deviations. 

A bubble-tube is a round glass tube bent or ground so that 
its inside upper surface is circular on a longitudinal section. 
This is nearly filled with ether, the remaining space being 
occupied with ether-vapor, which forms the bubble. This 
tube is usually graduated to assist in determining the exact 
position of the bubble in the tube. If the tube has been 
ground to a perfect circular longitudinal section, then a longi- 
tudinal line tangent to this inner surface at the centre of the air- 
bubble is a level line y in whatever part of the tube the bubble 
may lie. If this were not a level line, the centre of gravity of 
the bubble would not occupy its highest possible position and 
would move until it did. Since the position of the centre of 
a bubble in a tube is determined by reading the position of its 
ends and taking the mean, it is necessary that the arc shall be 
of uniform curvature— that is, circular. 

A line tangent to the inner surface of the bubble-tube at 
its centre, as defined by the graduations (or another line parallel 
to it) is called the axis of the bubble. When the bubble is in 
the centre of the tube, therefore, its axis is horizontal. 

Proposition I. If a bubble-tube be rigidly attached to a 
frame, and if this frame be reversed on two supports lying in 
the. vertical plane through the bubble-axis, the supporting 
points are level when the bubble occupies the same portion of 
the tube in both positions of frame, whether this be the centre 

* In the northern portion of the United States, in the vicinity of the Great 
Lakes, deviations of the plumb-line (Clarke's Spheroid being used) have been 
found as great as 10 or 12 seconds of arc. See Primary Triangulation of the 
U. S. Lake Survey. 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 57 



or not ; providing, of course, that the points of support on the 
frame were identical in the two cases. 

For, the tangent horizontal lines being identical in the two 
positions of the bubble, the vertical distances from this line to 
the points of support must be equal, otherwise the direct and 
reversed positions would not give identical tangent lines. The 
points of support are therefore in a horizontal line. 

Proposition II. If a bubble-tube be revolved about an axis 
in such a way that the bubble keeps a constant position in the 
tube, the axis of revolution is vertical. 

For, since the bubble-tube maintains a constant inclination 
to the horizon (this inclination being zero when the bubble is 
in the centre), the plane of motion can have no vertical com- 
ponent, and, therefore, the axis of revolution must be vertical. 

Cor. I. Similarly we may say that if a bubble-tube be re- 
volved 180 about an axis, and if the bubble have the same 
reading in the two positions, then the plane of revolution has 
no vertical component in the direction of the bubble-axis, and 
therefore the axis of revolution lies in a vertical plane at right 
angles to the bubble-axis. If the same test be made for any 
other two horizontal positions 180 apart (preferably 90 from 
first position) and the bubble have the same reading in the 
two cases, then the axis of revolution lies in a vertical plane at 
right angles to these new positions of bubble-axis, and there- 
fore it lies at the intersection of these two vertical planes, or it 
is vertical. If two bubble-tubes (not parallel to each other 
and preferably at right angles) be rigidly attached to a frame 
that revolves about an axis, and if each bubble has the same 
reading in two positions of frame 180 apart, the axis of revo- 
lution is vertical, even though the two bubbles do not read 
alike nor either is at the middle of its tube. 

Cor. 2. In all cases where a bubble-tube has been shifted 
180 in the same supports, or axis, the angular difference 
between the two positions of the bubble is twice the angular 



58 SURVEYING. 



deviation of the supports from a horizontal, or of the axis from 
a vertical. 

61. The Accurate Measurement of Small Vertical An- 
gles is accomplished by means of the bubble with greater read- 
iness and precision than by any other device known. For this 
purpose the bubble should be ground accurately to the arc of 
a circle with a long radius, and uniformly graduated. Then a 
given bubble-movement in any part of the tube corresponds to 
a known angular change, when the angular value of a move- 
ment of one division in the graduated scale has been deter- 
mined. These graduations are usually made on the top of the 
glass tube. To measure a small angle by means of the bubble, 
read the two ends of the bubble to divisions and tenths, and 
take the one half-difference of end readings.* Shift the bubble 
a given amount and read both ends again, taking one half the 
difference. The difference of these two results in divisions of 
the scale, multiplied by the angular value of one division on 
the scale, is the vertical angle through which the tube was 
shifted. 

62. The Angular Value of One Division of the Bubble 
may be found in various ways. 

(a) By a telescopic line of sight. Attach the bubble-tube rig- 
idly to a mounted telescope, putting the bubble-axis in the plane 
of the telescope. Measure off a convenient base-line on level 
ground of from 200 to 500 feet. Set the telescope at one end 
of this base, and hold a rod vertically at the other. Bring the 
bubble near one end of its tube by moving the telescope verti- 
cally, and read the two ends. Read the height of the cross- 
wires on the rod. Bring the bubble near the other end of tube 
and read both the bubble and rod. Repeat many times. Re- 
duce the work by taking the half-difference of the two end 

* Bubbles are read from the middle outwards towards the ends. Then the 
half-difference of end readings is the distance of the centre of the bubble from 
the centre of the scale. 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 59 

readings in each case, thus giving the distance of the centre of 
the bubble from the centre of tube for each position. Take 
the mean of these results for each set of end readings sepa- 
rately. If these mean results were for opposite ends of the 
tube, add them together and this gives the average movement 
of bubble. Similarly take the mean of the upper readings and 
the mean of the lower readings on the rod, and take the differ- 
ence, and this is the average movement of the line of sight. 
Calling the bubble-movement in divisions of scale D, the move- 
ment on the rod, in feet, R, and the length of the base, in feet, 
B, we would have, in seconds of arc, 

angular value of I div. of bubble = pr> . — — . 
& BD sin 1 

(b) By a large vertical circle. Mount the bubble rigidly 
upon the circle, having its axis parallel to the plane of the 
circle. Move the bubble from end to end of tube, as before, 
reading the corresponding angular changes directly upon the 
circle. Divide the mean angular movement by the mean 
movement of bubble. 

This requires a large circle with micrometer attachments, 
such as is used on astronomical instruments. 

(c) By a level trier. This consists of a beam hinged at one 
end and moved vertically by means of a micrometer screw at 
the other. The bubble-tube is placed upon the beam, and the 
bubble moved back and forth by means of the screw, each 
revolution of which gives a known angular movement to the 
beam. 

63. General Considerations. — A bubble is sensitive direct- 
ly as the length of the radius of curvature, or indirectly as its 
rate of curvature. It is also sensitive in proportion to its 
length, a long bubble* settling much more quickly and ac- 

*This refers to the length of the air-bubble itself, and not to the glass tube. 



6o SURVEYING. 



curately than a short one. Some bubble-tubes have a cham- 
ber at one end connected with the main space by a small hole 
through the bottom of the dividing partition. This enables 
the length of the bubble to be under control. As either ex- 
pands and contracts very largely with temperature, the bubble 
is apt to be too long in winter and too short in summer if the 
chamber is not used. The bubble-tube should not be rigidly 
confined by metallic fastenings about its centre, if the value of 
one division is significant, as the changes of temperature will 
change its curvature. Bubble-tubes, or level-vials as they are 
often called, may be sealed by glass stoppers set in a glue 
made by dissolving isinglass -in hot water, and covering with 
gold-beater's skin set with the same glue, the whole varnished 
over when dry. 

THE ENGINEER'S LEVEL. 

64. The Engineer's Level consists of a telescopic horizon- 
tal line of sight joined to a spirit-level, the whole properly, 
supported and revolving on a vertical axis. Such an instru- 
ment is shown in Fig. 12. The vertical parts of the frame 
which support the telescope are called wyes, and the cylindri- 
cal bearings on the telescope-tube are called the pivot-rings. 
The telescope can be lifted out of the wyes by loosening the 
clips over the rings, these being held by the small pins 
attached to strings and shown in the cut. A clamp and 
tangent-screw are connected with the axis for holding it to a 
given pointing or for moving it horizontally while clamped. 
The attached bubble enables the line of sight in the telescope 
to be brought into a horizontal position. 

The construction of the instrument is best shown by the 
sectional view given in Fig. 13. 

The objective is a compound lens, the two parts having 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 6 1 



different refractive powers in order that the image may be 
flat. A simple lens gives a spherical image. The image is 




formed at the plane of the cross-wires, which are attached to the 
reticule held in place by the capstan-screws shown in the cut. The 



62 



SURVEYING. 




line of sight, or line of collimation, is the line joining the two 
corresponding points in object and image with which the 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 63 

intersection of the cross-wires coincides.* Evidently this 
line of sight may lie anywhere in the field of view within the 
limits of movement of the reticule. The eye-piece serves only 
to magnify the image, and sometimes to invert it, as is the 
case in the sectional view of Fig. 13. The image itself is 
always inverted ; and if this be examined by an eye-piece of 
two lenses, which simply magnifies but does not invert, the 
object is seen in an inverted position. If four lenses are used ■ 
in the eye-piece, it re-inverts the image so that the object is 
seen erect. This results in a loss of light and of distinctness. 

ADJUSTMENTS OF THE LEVEL. 

65. To make the Line of Sight parallel to the Axis of 
the Bubble. 

First, or Indirect, Method. — This method rests on the 
proposition that if two lines are parallel to a third line, they are 
parallel to each other. This method is indirect, but the 
manipulations are readily performed. It is the usual method, 
and is frequently given as two separate adjustments. 

First, bring the line of sight to coincide with the centres of 
the pivot-rings by revolving the telescope, bottom side up, in 
the wyes, and adjusting the reticule until the intersection of 
the wires remains on a fixed point of the image. f If the 

* More correctly, it is the line joining the inner principal point of the objec- 
tive with that point of the image covered by the intersection of the cross-wires. 
See Fig. 61, and note to same. 

f The optical axis of a lens is the line joining the centres of the true spherical 
surfaces bounding it. If this axis is not coincident with the axis of the tele- 
scope, or rings, owing to an erroneous adjustment of the objective slide by the 
screws near the centre of the telescope tube, Fig. 13, or the improper setting of 
the lens in its case, then the image will be shifted laterally a small amount equal 
to the lateral deviation of the two " principal points" of the lens from each other. 
In this case the image itself will appear to rotate as the telescope is revolved. 
If now the centre of the cross-wires be moved so as to remain on a fixed portion 
of the image, it no longer occupies the axis of the telescope, but the line of sight 
is now parallel to this axis, so that this adjustment still accomplishes all that is 



64 SURVEYING. 



instrument gives an erect view of the object, there is one 
inversion between the wires and the eye, and therefore the 
reticule must be moved in the direction of and one half the 
amount of its apparent displacement. If the view is inverted, 
there is no inversion between wires and eye, and therefore its 
apparent is its true displacement. 

Second, make the axis of the bubble-tube parallel to the 
bottoms of the rings by reversing the telescope end for end 
in the wyes and adjusting the bubble until it remains in the 
centre of the tube for the two positions. The telescope 
should be removed and replaced with great care so as not to 
disturb the relative elevation of the wyes by any jar or shock. 
The axis, of course, should be clamped to prevent any hori- 
zontal motion in making either part of this adjustment. 
' This method is based on an assumption which may or may 
not be true. It is that the pivot-rings are of the same size, 
and therefore the lines joining their centres and bottoms are 
parallel. 

To find the relative size of the pivot-rings, use a striding- 
level resting on the two pivot-rings and read in reversed posi- 
tions. Then change the rings in their supports and read the 
level again in reversed positions. To reduce the notes, the 
value of one division of the striding-level must be known.* 

The objective is always properly centred and adjusted when 
the instrument leaves the maker's hands ; but it is apt to 
become loose in its frame, and this frame also loosens in the 
telescope-tube. If the glass is loose in its frame, unscrew it 
from the telescope-tube and screw up the tightening band 



desired. Or, the objective may have its optical axis coincident with that of the 
telescope and the optical axis of the eye-piece not parallel to that of the objec- 
tive, and this will cause the image and wires to appear to rotate together — when 
the telescope is revolved. This need cause no error in the work, but should be 
adjusted by the screws shown just back of the capstan screws, Fig. 13. 
* See adjustments in Precise Levelling, Chap. XIV. 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 6$ 

from the rear side. Do not take the glasses apart under any 
circumstances, for they are ground for a given relative position 
and would not be true for any other. A loose objective is a 
fatal defect in a levelling-instrument and must be constantly 
guarded against. 

Second, or Direct, Method. — This consists in adjusting the 
bubble directly to the line of sight, whether this be in the cen- 
tre of the pivot-rings or not. It is sometimes called the 
" peg adjustment." Drive two pegs on nearly level ground 
about 200 feet apart. Set the level about eight or ten inches 
from one of them, or so that when the rod is held upon it in a 
vertical position the eye end of the telescope will swing about a 
half inch from its face. Turn the eye end of the telescope upon 
the graduated face of the rod, the bubble being in the middle of 
its tube; look through the object end and set a pencil-point on 
the rod at the centre of the small field of view, which should 
be from -J- to J inch in diameter. Read the elevation of this 
point, which we will call a. Hold the same rod on the distant 
peg and, with the bubble in the middle, set the target on the 
line of sight, and call this reading b. Now carry the instru- 
ment to the distant peg, set it near it, read the elevation of the 
instrument as before, which reading we will call a' ; carry the 
rod to the first peg and set the target on the line of sight, giv- 
ing the reading b r . If the line o£ n sight had been parallel to the 
axis of the bubble in each case, it would have been horizontal 
when the bubble was in the middle of the tube, and hence the 
difference between the a and b readings in each case would 
have been the difference of elevation of the pegs.* We 
should therefore have had 

a-b = a'-b f (1) 

*This assumption neglects the effect of the earth's curvature. This is 
eight inches to one mile, and is proportional to the square of the distance. For 
200 feet it would be about 0.001 of a foot, and twice this, or 0.002 of a foot, is 
the error made in the above assumption. 

5 



66 SURVEYING. 



If the line of sight was not parallel to the axis of the bub- 
ble, however, then the differences of elevation of the two pegs, 
as obtained by the two sets of observations, are not equal, and 
we should have 

{a-b)-{b' -a f ) = d (2) 

Now d is twice the deviation of the line of sight from the 
bubble-axis for the given distance. (Let the student construct 
a figure and show this.) If, therefore, the target be moved 
up or down as the case may be, a distance equal to \d, then 
the line of sight may be brought to this position by the 
levelling-screws, and the bubble adjusted to bring it to the 
middle, or else the instrument may be left undisturbed with 
the bubble in the middle, and the line of sight adjusted to 
read upon the target by moving the reticule. The significant 
fact is that by moving the target \d from its last position a 
true horizontal line is established, and either the bubble or the 
line of sight can be adjusted to it after the other has been 
brought into a horizontal position by means of the levelling- 
screws. Equation (2) may be written 

(a + a')-(6 + P)=J; (3) 

from which it may be seen at once that the line of sight 
inclines down when d is positive, and up when d is negative. 
We may therefore have for setting the target the following 

Rule: Add together the two heights of instrument and the 
two rod readings, subtract the latter from the former, and take 
one half the remainder. Move the target by this amount from 
the b reading, up when positive and down when negative. It is 
then in a horizontal line with the cross-wires of the instrument. 

It will be noted that no distances are measured in the above 
method as is usually prescribed in peg-adjustments. After 
adjusting either the line of sight or the bubble at the second 
peg, return to the first peg, read height of instrument again, 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 67 

and then read the rod on the second peg for a check. See if 
this new value of (a — b) agrees with the adjusted value of 
{b' — a'). If not, adjust again. 

This method is independent of the relative size of the pivot- 
rings and of the condition of the objective. (The objective 
must have a fixed condition or no adjustment is worth any- 
thing.) Although the essential relation of parallelism is ob- 
tained between the line of sight and the bubble, it must not 
be expected that the telescope can be reversed in the wyes or 
revolved 180 about its axis without both these auxiliary 
adjustments appearing to be in error. For inasmuch as these 
two lines have been made parallel without reference to the 
axis of the telescope or to the bottoms of the rings, they 
probably are not parallel to either of these. If the first meth- 
od is used and the adjustment made, it should stand the test 
of the second (the necessary assumptions being true), but if 
adjusted by the second method it should not be expected to 
stand the test of the first method. At the same time the 
second method is absolute, while the first is based on assump- 
tions that are often untrue. This adjustment should be exam- 
ined every day in actual practice. 

66. To bring the Bubble-axis into the Vertical Plane 
through the Axis of the Telescope. — Turn the telescope 
slightly back and forth in the wyes, and note the action of the 
bubble. If it remains in the centre the adjustment is correct. 
If not, move one end of the bubble by means of the lateral 
adjusting-screws. If this adjustment is very much in error it 
should be made approximately right before going on with the 
preceding adjustment. 

67. To make the Axis of the Wyes perpendicular to 
the Vertical Axis of the Instrument. — This is to enable the 
telescope to be revolved horizontally without re-levelling. 
Level the instrument in one position. Revolve 180 horizon- 
tally, and correct one half the movement of the bubble by the 



68 SURVEYING. 



wye-adjustment and the other half by the levelling-screws. 
Repeat for a check. 

68. Relative Importance of Adjustments. — The first 
adjustment is by far the most important. The second can 
only enter in the work when the telescope is revolved slightly 
from its true position in the wyes. Most modern levels have 
some device for holding the telescope in its proper position 
when in use. This position is such as brings the horizontal 
wire truly horizontal. The last adjustment given is only a 
matter of convenience. It saves stopping to relevel after re- 
volving the telescope. It does not affect the accuracy of the 
work appreciably. It is absolutely essential, however, that the 
line of sight should be truly horizontal when the bubble is in 
the middle of the tube, or reads zero, and this makes the first 
adjustment here given of such vital consequence. 

69. Focussing and Parallax. — The eye-piece serves to 
give a distinct and magnified view of the image. It also inverts 
the image in all instruments where the object is seen in an 
erect position. Since the magnifying power of the eye-piece 
is large, its focal range of distinct vision is very small, depend- 
ing on its magnifying power. With the ordinary field-instru- 
ments it is about one sixteenth of an inch. Both the virtual 
image, as formed by the objective, and the cross-wires, should 
lie in the focus of the eye-piece. They should therefore lie in 
the same plane. Now the virtual image may be moved back 
and forth by moving the objective in or out, but the plane of 
the cross-wires is fixed. If the two are brought into the same 
plane, therefore, the image must be brought upon the wires. 
To accomplish this, first focus the eye-piece on the wires so that 
they appear most distinct. In doing this there should be no 
image visible, so that either the objective is thrown out of 
focus or the telescope is turned to the sky. The eye-piece is 
most accurately focussed by finding its inner and outer limits 
for distinct vision of the wires, and then setting it at the mean 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 69 

position. The objective may now be moved until the image 
also comes into focus. This will have to be done for each 
pointing if the 'distances are different. If the image is not 
brought into exact coincidence with the cross-hairs, these will 
seem to move slightly on the image as the eye is moved behind 
the eye-piece. This angular displacement of the wires on the 
image is called parallax, and can only occur when they are not 
in the same plane. It is removed by refocussing the object- 
ive, thus moving the image, until there is no perceptible rela- 
tive movement of wires and image as the eye is shifted, when 
they are practically in coincidence. If there is parallax, the 
reading may be in error by its maximum angular amount. If 
the eye were always held at the centre of the eye-piece there 
would be no parallax, and it is to accomplish this that the eye- 
piece is covered by a shield with a small hole in its centre. 
Still, the slight movement of the eye thus allowed is sufficient 
to cause some parallactic error if the wires and image are not 
practically coincident. When the eye-piece is once adjusted to 
distinct vision on the cross-wires it requires no further atten- 
tion so long as the instrument is used by the same person. 
Another person, having eyes of a different focal range, would 
have to readjust the eye-piece. The eye-piece adjustment, 
therefore, is personal, and is made once for all for a given indi- 
vidual; while the objective adjustment depends on the dis- 
tance of the object from the instrument, is made for each 
pointing, and is considered perfect when the parallax is re- 
moved.* 



* This discussion is worded for an erecting telescope, where the objective 
moves. In an inverting instrument the eye-piece and reticule may move together 
in the telescope while the objective remains fixed. Here the image takes differ 
ent positions in the telescope-tube, as the distances vary, and the cross-wires 
are moved to suit. There is also a motion of the eye-piece with reference to 
the wires, and this is the eye-piece adjustment ; while the movement of both 
together is what is called the objective adjustment in the above discussion. 



7o 



SURVEYING. 



6r= 



ll 



4l 




SLrf 







Fig. 14. 



Fig. 15. 



THE LEVELLING-ROD. 

70. The Levelling-rod is used to 
measure the vertical distance from the 
line of sight down to the turning-point 
or bench-mark. There are two general 
classes, Self-reading, or Speaking, and 
Target Rods. 

A Self -reading, or Speaking, Rod is 
one so graduated as to enable the ob- 
server to note at once the reading of the 
point which lies in the line of sight, this 
reading being in all cases the distance 
to the bottom of the rod. The rod- 
man here has nothing to do but to hold 
the rod vertical. The observer notes 
and records the reading. 

A Target-rod is furnished with a 
sliding target moved by the rodman in 
response to signals from the observer 
until it accurately coincides with the 
line of sight. Its position is then read 
with great accuracy by means of a ver- 
nier scale. 

Fig. 14 is one form of self-reading 
rod which is also fitted with a target. 
This is called the Philadelphia rod. 
Fig. 15 is the New York rod and is not 
self-reading. It is the standard target- 
rod used in this country. The one 
here shown is in three sections, whereas 
those in common use are in two parts 
only. 

Various other patterns of self-read- 
ing rods are used. For rough work a 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 7 1 

twelve- or fourteen-foot rod, 2 inches wide and \\ inches thick, 
painted and fitted with an iron or brass shoe at bottom, gradu- 
ated to hundredths of a foot, will be found very efficient. The 
graduations should be so distinct that they can be read through 
the telescope at a distance of five or six hundred feet. 

THE USE OF THE LEVEL. 

71. The Level is used — 

(a) To find the relative elevation of points a considerable 
distance apart. 

(&) To obtain the profile of a line. 

(c) To establish a grade. 

These objects may be more or less intermingled in any 
given piece of work. Whatever may be the ultimate object of 
the work, however, the immediate object for any given setting 
of the instrument is to find how much higher or lower a certain 
forward, or unknown, point is than a certain other back, or 
known, point. Thus, the rod being held on the known point, 
the line of sight is turned upon it and the rod-reading gives at 
once the height of instrument above that point. If the rod be 
now held on the forward, or unknown, point, and the line of 
sight turned upon it, this rod-reading gives the distance of that 
point below the line of sight. The reading on the known 
point is called the back-sight, and that on the unknown point 
is called the fore-sight. If the elevation of the known point 
be given, we find the elevation of the line of sight by adding 
the rod-reading at that point. By subtracting from this eleva- 
tion the reading on the unknown point, the elevation of that 
point is obtained. Thus we have found the relative elevations 
of the two points by referring them both to the horizontal 
plane through the instrument. Since the back-sight reading 
gives the elevation of the instrument, and since this is always 
greater than the elevation of that point, it follows that the 
back-sight reading is essentially positive. For a similar reason 



SURVEYING. 



the fore-sight reading is essentially negative, since any point 
on which the rod is held is lower than the line of sight. 
It will also be seen that there can be but one back-sight (un- 
less the height of the instrument is to be found from readings 
on several known points, and the mean taken), while there can 
be any number of fore-sights from one instrument position. 
Thus, the height of the instrument having been determined, 
the elevations of any number of points, in any direction, may 
be determined by referring them all to the horizontal plane 
through the instrument, whose elevation has been obtained by 
the single back-sight reading. It is also important to remem- 
ber that the terms " back-sight" and " fore-sight" have no 
reference to directions or points of the compass, but they do 
have a rational significance when we think of the work pro- 
ceeding from the known point to the unknown point or points. 
Thus, we refer back to the known point for height of instru- 
ment, and then transfer this knowledge forward to the points 
whose elevations we wish to find. 

DIFFERENTIAL LEVELLING. 

72. Differential Levelling consists in finding the differ- 
ence of elevation of points a considerable distance apart. The 
elevation of the first point being known or assumed, the differ- 
ence of elevation between this and any other point is found 
and added algebraically, thus giving the elevation of the second 
point. The " plane of reference" is the surface of zero-eleva- 
tion and is generally called the "datum plane." This is not 
really a plane but a level surface, according to the definition 
given in art. 59. It is, however, universally denominated the 
"plane of reference," "datum plane," or simply "datum." 
The problem, then, is to find the difference of elevation between 
two distant points. If the points were near together and had 
not too great a difference of elevation, a single setting of the 
instrument would be sufficient. If they are too far apart for 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 73 

this, either in distance or in elevation, then more than one 
setting of the instrument must be made. In this case the 
intervening points occupied by the rod are called turning-points, 
the terminal points being called bench-marks. The successive 
differences of elevation of these turning-points is determined 
by setting the level equally distant from them, and so they 
serve to divide up the total distance between terminal points 
into a series of short spaces, each of which can be covered by 
a single setting of the instrument. The successive differences 
of elevation of turning-points being found, their algebraic sum 
would be the difference of elevation of the terminal points, or 
bench-marks. But since all the back-sights are essentially 
positive and all the fore-sights are essentially negative, we may 
at once add all the back-sights together and all the fore-sights 
together, and take the difference of the sums. This is the 
difference of elevation between terminal points, and has the 
sign of the larger sum, the back-sights being positive and the 
fore-sights negative. This difference of elevation added alge- 
braically to the elevation of the initial point gives the elevation 
of the final point. Evidently the route travelled in passing 
from one bench-mark to another is of no consequence so long 
as the true difference of elevation is obtained. 

73. Length of Sights. — Where the ground is nearly level 
it is desirable to make the length of sights (distance from 
instrument to rod) as long as practicable, in order to increase 
the rate of progress. For the best work this distance may be 
from 100 to 300 feet, according to the state of the atmosphere. 
When the air and ground differ greatly in temperature there 
result innumerable little upward and downward currents of 
air, the upward being warmer than the downward currents. 
The warmer air is more rarefied than the colder, and thus a 
ray of light passing from the rod to the instrument passes alter- 
nately through denser and rarer media, each change producing 
a slight refraction of the ray. This causes a peculiar tremulous 



74 SURVEYING. 



condition of the image in the telescope, so that it is difficult to 
determine just what part of it is covered by the cross-hairs. At 
such times the air is said to be " trembling" or " dancing" or 
" unsteady." It always occurs more or less in clear weather, 
owing to the earth then being hotter than the air, and it varies 
with the quality of the soil, cinders or gravel being very bad. 
When the air is in this condition the length of sights should 
be shortened. 

The back and fore sights for any setting of the instrument 
should always be equal in length. Levelling is the only kind of 
field-surveying wherein the instrumental errors may be thor- 
oughly eliminated without duplicating the observations. This 
may be done in levelling by making the back and fore sights 
of equal length. For, since the difference between back and 
fore sights is always the quantity used, it follows that if both 
are too large or too small by the same amount, the difference 
will be unchanged. If, when the bubble is in the middle of 
its tube, the line of sight is inclined upwards by a given small 
angle, then it has this relation to the horizontal on both fore 
and back sights, and if the lengths of sights were equal the fore 
and back rod-readings were equally in error. It is therefore 
very desirable that these sights should be made of equal length. 
Moreover, the effect of the earth's curvature is eliminated by 
so doing, however long the sights may be. There are other 
kinds of errors that are not eliminated by this means, but those 
that are eliminated are of sufficient importance to warrant 
great care to secure equal sights for each setting. If it is impos- 
sible to do this at any time, the inequality should be balanced 
off at the next one or two settings, by making them unequal 
in the opposite direction by the same amount. The equality 
of sights can be determined by pacing with sufficient accuracy. 

74. Bench-marks are fixed points of more or less perma- 
nent character whose elevations are determined and recorded 
for future reference. The general and particular location of a 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 75 

bench-mark should be so distinctly described that any one 
could find it from its description. Whenever the work is tem- 
porarily interrupted a temporary bench-mark is set, such as 
a substantial stake driven into the ground, or a spike in the 
root of a tree. The prime requisite of a good bench-mark is 
that it shall not change its elevation during the period in 
which it is to be used. If this period is not more than two or 
three years, a spike driven in the spreading root of a tree 
near the trunk and well above ground will serve. The wood 
should be trimmed away from it so as to leave a projecting 
spur that will not be overgrown. The tree itself should then 
be marked by notching or otherwise, and carefully located in 
the description. 

If the mark is to serve for from five to fifty years, stone or 
brick structures or natural rock should be selected. The water- 
tables, or corners of stone steps, of buildings, copings of founda- 
tion and retaining walls, piers and abutments of bridges, or 
copper bolts leaded in natural rock may serve. If artificial 
structures are chosen, those should be selected which have 
probably settled to a fixed position, and for this reason old 
structures are preferable to new ones. 

When stakes are used for temporary benches it is often 
advisable to set two or even three for a check. In this case 
the mean elevation is the elevation used. In starting from 
such a series of benches there would be as many back-sights 
for the first setting of the instrument as there were benches, 
the mean of which, added to the mean elevation of the benches, 
would give the height of instrument. In running a continuous 
line of levels it is advisable to set a bench-mark at least as 
often as one to the mile. 

75« The Record in differential levelling is very simple. 
The bubble always being put in the middle of the tube, and 
the rod-positions chosen equally distant from the instrument, 
the bubble-reading and the length of sights may be omitted 



7 6 



SURVEYING. 



from the record, unless some knowledge of the distance run is 
desired, when the length of sights may be inserted. 

Form of Record for Differential Levelling. 



No. of 
Station. 


Back-sights. 


Fore-sights. 


Elevation of 
Mean Benches. 


Remarks. 




3.426 

3.878 


4.879 
3-472 


96.301 
94.718 


B. S. on B. M. 31 
" " '* 31a 


I 
2 


3-652 
4.517 

3.216 




3 


4.361 

4.873 


F. S. on B. M. 32 

" " " 32a 




4617 






+11.385 


— 12.968 

+11.385 






- 1.583 





It will be seen that the mean of the readings on the two 
bench-marks was used in each case. The back-sights being 
essentially positive and the fore-sights essentially negative, 
these signs are prefixed to the sums, and the algebraic sum of 
these gives the elevation of the forward above or below the 
rear benches. This added to the elevation of the initial point 
gives the elevation of the final point. These points are the 
mean elevation of two bench-marks in the example given. 

76. The Field-work should be done with great care if 
the best results are to be obtained. The instrument should be 
adjusted every day, especially the parallelism of bubble-axis 
and line of sight. The instrument and rod should both be set 
in firm ground. An iron pin, about one inch square at top, 
six to eight inches long, and tapering to a point, should be 
used for the turning-point. A rope or leather noose should be 
passed through an eye at top to serve as a handle. To hold 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 77 

the rod upright the rodman should stand squarely behind it 
and keep it balanced on the pin. When the target is set and 
clamped the rodman reads it and records it on a paper he car- 
ries for the purpose. He then carries it to the observer, if it 
was a back-sight reading, or he awaits the coming of the ob- 
server if it was a fore-sight reading, when the observer also 
reads it and records it in his note-book. The rodman then 
calls off his reading, and the observer notes its agreement with 
his recorded reading. In this way two wholly independent 
readings are obtained and any erroneous reading corrected. 
Errors of one foot or one tenth are not very uncommon in 
reading target rods. The rodman should be especially careful 
to protect the turning-point from all disturbances between the 
forward and back readings upon it. The observer must not 
only obtain an accurate bisection on the target, but he must 
know that the bubble is accurately in the centre of the tube 
when this bisection is obtained. When the observer walks for- 
ward to set his instrument he counts his paces, and takes as 
long a sight as the nature of the ground or the condition of 
the atmosphere will allow. When the rodman comes up he 
counts his paces to the instrument and then goes the same dis- 
tance in advance. Thus the observer controls the length of 
sights, making them whatever he likes ; and it is the business 
of the rodman to see that the back- and fore-sight for every 
instrument-station are equal. 

PROFILE LEVELLING. 

77. In Profile Levelling the object is to obtain a profile of 
the surface of the ground on certain established lines. Here 
both the distances from, and the elevation above, some fixed 
initial point must be obtained. When the line is laid out 
stakes are usually driven every hundred feet, these positions 
being obtained by a chain or tape. It is now the business of 
the leveller to obtain the elevation of the ground at each of 



?8 SURVEYING. 



these stakes, and at as many other intermediate points as may 
be necessary to enable him to draw a fairly accurate profile of 
the ground. The ioo-foot stakes are usually numbered, and 
these numbers are entered on the level record. The inter- 
mediate points are called pluses. Thus, a point 40 feet beyond 
the twenty-fifth 100-foot stake is called 25 -f- 40, being really 
2540 feet from the initial point. It is evident that no plus- 
distance can be more than 100 feet, and these are usually paced 
by the rodman. The intermediate points are selected with 
reference to their value in determining the profile. These are 
points where the slope changes, being mostly maximum and 
minimum points, or the tops of ridges and bottoms of hollows. 
Turning-points are selected at proper distances, depending on 
the accuracy required, and these may or may not be points in 
the line whose profile is desired. The levelling-instrument also 
is not set on line, if it is found more convenient to set it off the 
line. 

In profile levelling, since absolute elevations with reference 
to the datum-plane are to be obtained from every instrument- 
position, it is necessary to find the height of instrument above 
datum for every setting, and from this height of instrument, 
obtained by a single, back-sight reading on the last turning- 
point, the elevations of any number of points are found by sub- 
tracting the readings upon them. 

78. The Record in profile levelling is much more elaborate 
than in differential levelling. The following form is considered 
very convenient for profile-work where the line has been laid 
out and 100-foot stakes set :* 

*This sample page was contributed to Engineering News in June, 1879, and 
the form of record is credited to Mr. E. S. Walters, a railroad engineer of large 
experience. 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 79 





Guatemala and 


Honolulu Railroad. Feb. 30, 1876 


• 


B. S. 


El. of T. P. 
and B. M. 


F. S. 


H.I. 


I. S. 


S. E. 


Sta. 


Remarks. 


10.552 


195-497 




206 . 049 






B. M. 

188 + 44 
189 
190 

+ 30 

191 
+ 20 

192 

T. P. 

193 
+ 5o 
B. M. 

104 

195 
T. P. 

196 

197 
+ 60 
+ 65 

198 

199 
T. P. 

+ 35 
200 
201 




9-32 
11. 41 

7.01 
2.07 
1.62 
0.38 
82 


196.73 
194.64 
199.04 
203 . 98 
204.43 
205.67 
205.13 
































































0.5I5 


202.797 


3-252 


203.312 




3.10 

2.70 

5.264 

8.20 

9-35 


200.21 
200.61 

I95. II 
I93.96 
















198.048 


























3-4H 


I94.840 


8.472 


198.251 




4.28 
5.06 
7.20 
10.60 
7.00 
5-46 


193-97 
I93-I9 
I9I-05 
187.65 

I9I-25 
I92.79 




















































9.527 


I95-083 3.168 


204.610 




10.25 
8.62 
6.04 


I94.36 

195.99 
198.57 












24.005 
14.892 




14.892 
























9- TI 3 



























In the above headings, B. S. denotes back-sight; F. S., fore-sight ; I. S., 
intermediate sight ; H. I., height of instrument ; T. P., turning-point ; B. M., 
bench-mark ; S. E.. surface-elevation ; Sta., station. 



It will be noted that there is but one back-sight and one 
height of instrument for each setting. The back-sight and 
fore-sight readings from the same instrument-station are not 
found here on the same line, as in differential levelling, but the 
fore- and back-readings on the same turning-point are on the 
same line. Thus, the rod was first read on the bench-mark 
whose elevation was known to be 195.497 feet above datum. 
The reading on this bench was 10.552, thus giving a height of 



80 SURVEYING. 



instrument of 206.049. This is marked B. M. in the station 
column, and evidently has but one reading upon it in starling 
the work from it. A series of intermediate sights are then 
taken at various 100-foot stakes and pluses, the readings on 
which, when subtracted from the H. I., give the surface-eleva- 
tions at those points. When the work has progressed as far in 
front of the instrument as the B. M. was back of it, a turning- 
point is set, and the reading upon it recorded in the column of 
fore-sights. This reading was 3.252, which, subtracted from the 
H. I. 206.049, gives 202.797 as the elevation of the turning- 
point. The instrument is now moved forward and a back- 
sight reading taken upon this T. P. of 0.515, which added to 
202.797 gives 203.312 as the new H. I. At this setting a new 
bench was established by taking an intermediate sight upon it 
of 5.264, and writing the elevation in the B. M. column instead 
of in the S. E. column. The readings on bench-marks and 
turning-points are made to thousandths, while the intermediate 
sights for surface-elevation are read only to hundredths of a 
foot. The last height of instrument is checked by adding the 
back-sights and fore-sights, taking the difference and applying 
it to the elevation of the initial point with its proper sign, re- 
membering that back-sights are positive and fore-sights negative. 
The profile is now constructed by the data found in the S. E. 
and Sta. columns, these being adjacent to each other. One 
of the great merits of this form of record is that wherever 
it is necessary to combine any two numbers by addition or 
subtraction, they are found in adjacent columns. In construct- 
ing the profile, some kind of profile or cross-section paper is 
used, and the horizontal scale made much smaller than the 
vertical. Thus, if the horizontal scale were 400 feet to the 
inch, the vertical scale might be 10 or 20 feet to the inch. 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 8 1 



LEVELLING FOR FIXING A GRADE. 

79. In fixing a grade the profile may be obtained and 
the grade marked upon it. The vertical distance between the 
surface-line and the grade-line, at any point is the depth of 
cut or fill at that point, and this may be marked on the line 
stakes at once, without the aid of the level or rod, if only the 
centre depths are desired, as in the case of a ditch or trench. 
If the sides are to have a required slope, however, the level 
and rod are necessary to fix the horizontal distance of the 
limiting or " slope" stakes from the centre stakes whenever 
the ground is not strictly a level surface. This operation is 
called " cross-sectioning," and is described in Chapter XIII., 
on Determination of Volumes. 

If the grade be known before the profile is determined, to- 
gether with the absolute elevation of the initial point, as is 
sometimes the case with ditches and trenches for pipe lines or 
sewers, then the depth of cut (or fill) may be at once deter- 
mined and marked on the line stakes when the profile is taken. 
The form of record might be the same as given above for pro- 
file levelling, with the addition of two columns after the " Sta- 
tion" column, one being Elevation of Grade, and the other Cut 
or Fill. The elevation of grade would be found for each pro- 
file point by adding if an up, and subtracting if a down, grade, 
the differences of elevation corresponding to the successive 
distances in the profile. The difference between the corre- 
sponding " surface-elevation" and " elevation of grade" would 
be trtfe cut or fill at each point, which could be at once taken 
out and marked on the line stake. 

THE HAND-LEVEL. 

80. Locke's Hand-level is a very convenient little instru- 
ment for rough work, such as is done on reconnaissance expedi- 
tions. It consists of a telescope with a bubble attached in 

6 



$2 



SURVEYING. 



such a way that the position of the bubble is seen by looking 
through the telescope. A horizontal line of sight is thus 
readily determined. It is supposed to be adjusted once for all. 




Fig. 16. 



EXERCISES WITH THE LEVEL. 

81. Adjust the bubble to the line of sight by the first, or indirect, method, 
and then test it by the second, or direct, method. If this second method does 
not show it to be in adjustment, where does the error lie ? 

82. Cause the line of sight and bubble-axis to make a considerable angle 
with each other (that is, put it badly out of adjustment in this particular), and 
level around a block or two, closing on the starting-point, being careful to 
make back and fore sights as nearly equal as possible. Of course the final 
elevation of the point should agree with the assumed initial elevation. The 
difference of these elevations is the error of closure of the level polygon. If 
the back and fore sights were exactly equal this should be zero, notwithstand- 
ing the erroneous adjustment. 

83. Put the instrument in accurate adjustment, and level over the same 
polygon as before, making the back and fore sights quite unequal, and note the 
error of closure. If the instrument were in exact adjustment and there were 
no errors of observation, should the error of closure be zero ? 

84. Range out a line on uneven ground about a half-mile in length, and set 
stakes every hundred feet. Let each student determine the profile indepen- 
dently. When all have finished, let them copy their profiles on the same piece 
of tracing-cloth, starting at a common point. The vertical scale should be 
large, so as to scatter the several profile lines sufficiently on the tracing. Each 
profile should be in a different color or character of line. 

85. Select a line on nearly level ground, about a half-mile in length. Estab- 
lish a substantial bench-mark at each end. Let each student determine the 
difference of elevation of these benches twice, running forward and back. See 
if the results are affected by the direction in which the line is run. 

If each student could do this several times some evidence would be ob- 
tained as to there being such a thing as " personal equation" in levelling ; that 
is, each person tending to always obtain results too high or too low. Why is 
it improbable that there could be any personal equation in levelling? 



ADTUSTMENT, USE, AND CARE OF INSTRUMENTS. ' 83 



CHAPTER IV. 

INSTRUMENTS FOR MEASURING ANGLES. 

THE TRANSIT. 

86. The Engineer's Transit is the most useful and 
universal of all surveying-instruments. Besides measuring 
horizontal and vertical angles it will read distances by means 
of stadia wires, determine bearings by means of the magnetic 
needle, do the work of a solar compass by means of a special 
attachment, and do levelling by means of a bubble attached 
to the telescope. It is therefore competent to perform all the 
kindb of service rendered by any of the instruments heretofore 
described, and is sometimes called the "universal instrument." 
A cut of this instrument is shown in Fig. 17. Fig. 18 is a 
sectional view through the axis of a transit of different 
manufacture. 

The telescope, needle-circle, and vernier plates are rigidly 
attached to the inner spindle which turns in the socket Ci 
Fig. 18. This portion of the instrument is called the alidade, 
as it is the part to which the line of sight is attached. The 
socket C carries the horizontal limb, shown at B, and may 
itself revolve in the outer socket attached to the levelling-head. 
Either or both of these connections may be made rigid by 
means of proper clamping devices. If the horizontal limb B 
be clamped rigidly to the levelling-head and the alidade spindle 
be allowed to revolve, then horizontal angles may be read by 
noting the vernier-readings on the fixed horizontal limb for 
the different pointings of telescope. If the horizontal limb 
itself be set and clamped so that one of the verniers reads zero 




Fig. 17. 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 85 



when the telescope is on the meridian, then for any other 
pointing of the telescope the reading of this same vernier 
gives the true azimuth of the line. It is necessary, therefore, 
to have two independent movements of telescope and horizon- 
tal limb on the same vertical axis. The magnetic needle is 
shown at N. The plumb-line is attached at P; this should 
always be in the vertical line passing through the centre of 
the graduated horizontal circle. This will be the case when 




it is attached directly to the axis itself, for this must always 
be made vertical. 

The limb is graduated from zero to 360 , and sometimes 
with a second set of figures to 90 or 180 . There are two 
verniers reading on the horizontal limb 180 apart. Both the 
instruments shown in Figs. 17 and 18 have shifting centres, 
enabling the final adjustment of the instrument over a point 
to be made by moving it on the tripod-head. The telescope 
is shorter than those used in levelling-instruments in order 
that it may be revolved on its horizontal axis without having 
the standards too high. It is called a transit instrument on 



86 SURVEYING. 



account of this movement, which is similar to that of an 
astronomical transit used for observing the passage (transit) 
of stars across any portion of the celestial meridian. When 
the telescope is too long to be revolved in this way the instru- 
ment is called a theodolite. This is the only essential differ- 
ence between them.* The " plain transit " has neither a 
vertical circle nor a bubble attached to the telescope. 

ADJUSTMENTS OF THE T*RANSIT. 

87. The Adjustments of the Engineer's Transit are 

such as to cause (1) the instrument to revolve in a horizontal 
plane about a vertical axis, (2) the line of collimation to gen- 
erate a vertical plane through the instrument-axis when the 
telescope is revolved on its horizontal axis, (3) the axis of the 
telescope-bubble to be parallel to the line of collimation, thus 
enabling the instrument to do levelling, and (4) the vernier on 
the vertical circle so adjusted that its readings shall be the 
true altitude of the line of collimation. These four results are 
attained by making the following five adjustments : 

88. First. To make the Plane of the Plate-bubbles 
perpendicular to the Vertical Axis. — This adjustment is 
the same as with the compass. (One of the plate-bubbles is 
usually set on one pair of standards.) Bring both bubbles to 
the centre, revolve 180 , correct one half the movement on 
the levelling-screws and the other half by raising or lowering 
the adjustable end of the bubble-tube. Each bubble should 
be brought parallel to a set of opposite levelling-screws in 
making this adjustment, so that the correcting for one bubble 
does not throw the other out. When either bubble will main- 
tain a fixed position in its tube as the instrument is revolved 
horizontally, the axis of revolution is vertical. One bubble is 

*The first engineer's transit instrument was made by Wm. J. Young (now 
Young & Sons), Philadelphia, 1831. All American engineer's altitude-azimuth 
instruments are now made to revolve in this wa)r. 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 87 

therefore sufficient for making this axis vertical, but two are 
somewhat more convenient, especially for indicating when the 
axis has become inclined from unequal settling or expansion 
while in use. 

89. Second. To make the Line of Collimation perpen- 
dicular to the Horizontal Axis of the Telescope. — When 
this is done, the line of collimation will generate a plane when 
the telescope is revolved on its horizontal axis. If the line of 
collimation is not perpendicular to the horizontal axis, it gen- 
erates the surface of a cone when the telescope is revolved, the 
axis of the cone being the axis of revolution, and the apex 
being at the intersection of the line of collimation with this 
axis. 

Set the instrument on nearly level ground, where a view 
can "be had in opposite directions. Set the line of sight on a 
definite point a few hundred feet away. Revolve the telescope 
and set another point in the opposite direction. Revolve the 
alidade until the line of sight comes upon the first point. Re- 
volve the telescope again and fix a third point on the line of 
sight beside the second point set. Measure off one-fourth the 
distance between these two points from the last point set, and 
bring the line of sight to this position by moving the reticule 
laterally. This movement of the reticule is direct in an erect- 
ing instrument and reversed in an inverting instrument. 

The student should illustrate the correctness of this method 
by means of a figure. The four pointings were the intersec- 
tions of a diametral horizontal plane with the surfaces of the 
the two cones generated. These cones were pointed in oppo- 
site directions, but had one element in common, being the two 
pointings to the first point. The two opposite elements 
diverged by four times the difference between the semi-angle 
of the cone (subtended by the line of collimation and the axis 
of rotation) and 90 . 

90. Third. To make the Horizontal Axis of the Tele- 



88 SURVEYING. 



scope perpendicular to the Axis of the Instrument. — When 
this is done the former is horizontal when the latter is vertical, 
and, the second adjustment having been made, the line of sight 
will generate a vertical plane when the telescope is revolved. 

Set the instrument firmly and level it carefully. Suspend 
a plumb-line some 20 or 30 feet long, some 15 or 20 feet from 
the instrument. The weight should rest in a pail of water and 
the string should be hung from a rigid support. There should 
be no wind, and the cord should be small and smooth. A small 
fish-line is very good. Care must be exercised that the weight 
does not touch the bottom of the pail from the stretching of 
the cord. Set the line of sight carefully on the cord at top, 
the plate-bubbles indicating a strictly vertical instrument-axis. 
Clamp both horizontal motions and bring the telescope to read 
on the bottom portion of the cord. The cord is apt to swing 
to and fro slightly, but its mean position can be chosen. If the 
line of sight does not correspond to this mean position, raise 
or lower the adjustable end of the horizontal axis until this 
test shows the line of sight to revolve in a vertical plane. 
Constant attention must be given to the plate-bubbles to see 
that they do not indicate an inclined vertical axis. 

Or, two points nearly in a vertical line may be used, as the 
top and bottom of the vertical corner of a building. Set on 
the top point and revolve to the bottom point. Note the 
relation of the line of sight to this point. Revolve 180 about 
both vertical and horizontal axes, and set again on the top 
point. Lower the telescope again and read on the bottom 
point. If the telescope-axis of revolution is horizontal, the 
second pointing at bottom should coincide with the first. If 
not, adjust for one half the difference between these two 
bottom readings. 

It will be noted that the second and third adjustments are 
necessary to the accomplishment of the second result cited in 
art. 87. 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 89 

91. Fourth. To make the Axis of the Telescope-bub- 
ble parallel to the Line of Collimation. — This adjustment 
is performed by means of the " peg-adjustment," as described 
in art. 65, p. 65, second method. The height of the instrument 
may now be measured to the centre of the horizontal axis if it 
be found more convenient than sighting backwards through 
the telescope. When this adjustment is made the instrument 
is competent to do levelling the same as the levelling-instru- 
ment. The telescope is not quite so stable, however, in the 
transit because it is mounted on an axis instead of in two rigid 
wyes. 

92. Fifth. To make the Vernier of the Vertical Circle 
read Zero when the Line of Sight is Horizontal. — Having 
made the axis of the telescope-bubble parallel to the line of 
sight, bring this into the centre of its tube, and adjust the 
vernier of the vertical circle till it reads zero on the limb. If 
this vernier is not adjustable, the reading in this position is its 
index error. The line of sight might still be adjusted to the 
vernier by moving the reticule, and then adjusting the bubble 
to the line of sight. To do this use the " peg-adjustment " as 
described in art. 65, making the vertical circle read zero each 
time, and paying no attention to the telescope-bubble. Correct 
the line of sight by \ d. as given by Eq. (2), p. 66, by moving 
the reticule, and this should give a horizontal pointing for a 
zero-reading of the vertical circle. Then adjust the bubble to 
this reading by bringing it to the centre of the tube by means 
of the vertical motion at one end of the bubble-tube. If the 
reticule is disturbed after making the second adjustment, that 
adjustment should be tested again to see if it had been dis- 
turbed. 

93. Relative Importance of the Adjustments. — The first 
adjustment is important in all horizontal and vertical angular 
measurements. In measuring vertical angles the error may be 
the full amount of the deviation of the vertical axis from the 



90 SUR VE YING. 



vertical, and in measuring horizontal angles something very- 
much less than this. 

The second adjustment is more important in the running of 
a straight line by revolving the telescope than in any other kind 
of work, for here the error in the continuation of the line is 
twice the error of adjustment. It is also important in measur- 
ing horizontal angles between points not in the same horizontal 
plane. 

The third adjustment is most important in the measure- 
ment of horizontal angles between points not in the same hori- 
zontal plane, as in the determination of the azimuth of a line 
by an observation on a circumpolar star. 

The fourth and fifth adjustments are important only in 
levelling operations, either by reading the vertical angle or by 
the use of the bubble. 

INSTRUMENTAL CONDITIONS AFFECTING THE ACCURATE 
MEASUREMENT OF HORIZONTAL ANGLES.* 

94. Eccentricity. — This is of two kinds: (1) eccentricity of 
centres, and (2) eccentricity of verniers. If the axis of the coni- 
cal outer socket C, Fig. 18, is not exactly in the centre of the 
graduated limb B, then when the telescope with the vernier 
plates Fare revolved in this socket, the verniers will have an 
eccentric motion with reference to the graduated limb. If the 
line joining the zeros of the verniers passes through the axis 
of the socket, it is evident that there is but one position of 
these verniers which will give readings on the limb 180 apart, 
and that is when both centres lie in this diametral line. For 
all other positions of the verniers, one of them will read as 
much too large as the other does too small ; so that if the mean 

* For extended discussions of this subject, see Bauernfeind's " Vermessungs- 
kunde," § 144, vol. i., and Jordan's " Handbuch der Vermessungskunde," § 88, 
vol. i. Also translations from these, by Prof. Eisenmann, in Journal of the 
Association of Engineering Societies, vol. iv. p. 196. 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 9 1 



of the two vernier-readings be taken, this error from eccentric- 
ity would be eliminated. 

Eccentricity of verniers is due to their zeros not falling on 
a diametral line through the axis of the spindle ; in other 
words, they are not i8o° apart. This involves no error in 
measuring horizontal angles. It is convenient, however, to 
have the verniers read exactly 180 apart. In any case, read- 
ing of both verniers and taking the mean eliminates all errors 
from eccentricity. An eccentricity of centres of one one-thou- 
sandth of an inch would cause a maximum error of i'-oS" on a 
six-inch circle if but one vernier were read. It is not unusual 
for an instrument to have an eccentricity of centres of several 
times this amount, either from wear or from faulty construction, 
or both. The necessity for reading both verniers in all good 
work is therefore apparent. 

95. Inclination of Vertical Axis. — The horizontal angle 
between points at different elevations is obtained by measuring 
the horizontal angle subtended by two vertical planes passing 
through these points and the point of observation. These 
vertical planes are the planes described by the line of sight as 
the telescope is revolved. By this means the points may be 
said to be projected vertically on the horizontal plane and 
then the angle measured. If the vertical axis of the instru- 
ment is somewhat inclined, these projecting planes are not ver- 
tical, neither do they have the same inclination to the horizon 
on different parts of the limb. The projecting planes through 
two points will therefore neither be vertical nor equally in- 
clined to the horizon. The measured horizontal angle thus 
obtained will therefore be in error. The vertical axis is always 
inclined when the plate-bubbles are not in adjustment or when 
they do not show a level position. 

If the axis be inclined 5' from the vertical, and readings be 
taken on points 6o° apart, one being io° above and the other 
IO° below the horizon, the maximun error from this source 



9 2 SURVEYING. 



would be about i'. If the inclination in this case were i°, the 
maximum error would be 18'. This shows the importance of 
keeping the plate-levels in adjustment and of watching them 
during the progress of the work to see that they remain in the 
centre. 

96. Inclination of Horizontal Axis of Telescope. — 
This causes the plane generated by the line of sight to be in- 
clined from the vertical as much as the axis of revolution is 
from the horizontal. The projecting planes are therefore all 
equally inclined, and the resulting error in horizontal angle is 
a function of the difference of elevation of the two points. If 
one point is io° above and the other io° below the horizon, 
and if the inclination of the axis is 5', the resulting error in 
the measurement of the horizontal angle is i'-45". This error 
is not a function of the size of the horizontal angle, and would 
be the same for two points in the same vertical plane, the in- 
strument indicating a horizontal angle of \' 45" between them 
for the case here chosen. In making the adjustment of the 
horizontal axis by means of the plumb-line, if the line be 15 
feet distant and suspended 15 feet above the instrument, then 
the pointing to the top will have an altitude of 45 . In this, 
case the angular error made in bisecting the plumb-line will be 
the angular divergence of the axis of rotation from the hori- 
zontal. If the combined error of the two bisections be o. 05 in., 
the angular error in the adjustment will be i'. The adjust- 
ment may readily be made closer than this. 

Errors from this source are eliminated by revolving the 
telescope and reading the same angle in the reversed position. 
The mean of the two values will be independent of this error. 
If many measurements are made of one angle, there should be 
an equal number with telescope direct and reversed. 



The student should show by a figure how this elimination is effected by the 
reversal of the telescope. 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 93 

97. The Line of Collimation not being Perpendicular 
to the Horizontal Axis. — This causes the projecting planes 
to be conical surfaces, which become vertical on the horizon. 
Since the error of collimation is necessarily a small angle, thus 
causing the conical surface to be very nearly a plane, and since 
this surface is vertical on the horizon, the resulting error in 
measuring horizontal angles is very small unless the difference 
in the elevations of the points is very great. If the points are 
distant, as they always are in the accurate measurement of 
horizontal angles, then their angular elevation is necessarily 
small, so that this source of error is insignificant in this kind 
of work. When straight lines are prolonged by reversing the 
telescope, however, this adjustment becomes very important, 
for the error then enters the work with twice its angular 
amount. It is eliminated by revolving the alidade until the 
line of collimation, with telescope reversed, falls again on the 
rear point, and again revolving the telescope. The point now 
falls as far on one side of the true position as it before did on 
the other. The middle point lies therefore in the line pro- 
longed. 

Let the student illustrate by diagram. 

THE USE OF THE TRANSIT. 

98. To measure a Horizontal Angle. — Having centred 
the instrument over the vertex of the angle required, take a 
pointing to one of the points and clamp both alidade and 
limb. Make the final bisection by means of either tangent- 
screw. Read the two verniers, and record them, calling one 
the reading of vernier A and the other of vernier B. Loosen 
the alidade clamp and turn upon the second point, clamp, and 
set by the upper tangent-screw. Read both verniers again. 
Correct the readings of vernier A by half the difference be- 
tween the A and B readings in each case. The difference 
between these corrected readings is the value of the angle. 



94 SURVEYING. 



Be careful not to disturb the lower clamp- or tangent-screw 
after reading on the first point. If there are two abutting 
tangent-screws for the lower plate, be sure that both are 
snug, otherwise there may be some play here which would 
allow the limb to shift its position, in which case the true angle 
would not be obtained. If there is but a single tangent-screw 
working against a spring on the other side of the armature, 
as shown in Fig. 16, then there can be no lost motion unless 
the friction on the axis is greater than the spring can over- 
come, which should never be the case. 

Do not set the clamp-screws too tightly, as it strains and 
wears out the instrument unnecessarily. A very gentle press- 
ure is usually sufficient to prevent slipping. This caution 
applies equally well to all levelling-, adjusting-, and connecting- 
screws in the instrument. The young observer is generally 
inclined to set them up hard, as he would in heavy iron-work. 
It must be remembered that brass is a soft material, easily dis- 
torted and worn, and that the parts should be strained as little 
as possible to insure against movement in ordinary handling. 

The subject of measurement of horizontal angles is further 
discussed in Chapter XIV., on Geodetic Surveying. 

99. To measure a Vertical Angle.— Vertical angles are 
usually referred to the horizon, and are angles of elevation or 
depression above that plane. If the vernier on the vertical 
circle has been properly adjusted (or its index error determined 
in case it is not adjustable and the line of sight has not been 
adjusted to it), then the altitude of a point is obtained at once 
by turning the line of sight upon it and reading the vertical 
angle. Special attention must here be given to the bubble 
parallel to the vertical circle, for it is on this bubble that the 
accuracy of the result wholly depends. If there is but one 
vernier, it is designed to read both ways, as is shown in Figs. 5 
or 6, p. 19. In this case errors of eccentricity cannot be elim- 
inated. 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 95 

To eliminate errors of adjustment of the plate-bubbles and 
of the vernier on the vertical circle, revolve the alidade 180 , re- 
level, read the vertical angle again with telescope in a reversed 
position, and take the mean. This can only be done in case 
the vertical limb is a complete circle. In many instruments it is 
but a half-circle or less, in which case this elimination cannot 
be made. The accuracy of the adjustments alone can then 
be relied on, and these must be frequently tested. If the plate- 
bubble parallel to the vertical circle, the telescope-bubble, and 
the vernier of the vertical circle have all been once accurately 
adjusted, then when these bubbles are brought to a zero-read- 
ing the vertical circle should also read zero. This test can 
always be readily applied, and, though not an absolute check, 
it is a very good one, inasmuch as two of these three adjust- 
ments would have to be out by the same amount and in the 
same direction to still agree with the third. 

100. To run out a Straight Line. — The transit-instru- 
ment is especially adapted to the prolongation of straight lines, 
as long tangents on railroads, and yet it requires the most care- 
ful work and much repetition to run a line that approximates 
very closely to a straight line. 

Having determined the direction which the line is to take 
from the initial point, set accurately over this point, turn the 
telescope in the given direction, and set a second point at a 
convenient distance. These two points now determine the 
line, and it remains to prolong it indefinitely over such uneven 
ground as may lie in its course. The line, when established, 
is to be the trace of a vertical plane through the first two 
points on the surface of the ground. If the line of collimation 
always revolved in a vertical plane, and no errors were made 
in handling the instrument and in setting the points, the prob- 
lem would be easily solved, but we may safely say that the 
surface generated by the line of collimation never is a vertical 
plane. (The adjustments being never absolutely correct.) 



96 SUE VE YING. 



This surface is a cone whose axis is not strictly horizontal, for 
both the horizontal and vertical axes are somewhat inclined 
from their true positions." It remains then so to make the 
observations that all these errors of adjustment will be elimi- 
nated. The following programme accomplishes this : 

(1) Set accurately over the forward point, putting one pair 
of levelling-screws in the line. 

(2) Clamp the horizontal limb in any position. 

(3) Level carefully, and turn upon the rear point. 

(4) Relevel for the bubble that lies across the line. 

(5) Make the bisection on the rear point, revolve the tele- 
scope, and set a point in advance. This may be a tack in a 
stake set with great care by making the bisection on a pencil 
held vertically on the stake. 

(6) Unclamp the alidade and revolve it about the vertical 
axis till the telescope comes on the rear point. 

(7) Relevel for the cross bubble again. 

(8) Make the bisection on the rear point, revolve the tele- 
scope again, and set a second point in advance beside the first 
one. The mean of these two positions should lie in the verti- 
cal plane through the two established points, whatever may be 
its elevation, and regardless of small errors in the instrumental 
adjustments. For the reversals of the telescope and alidade 
eliminated the errors of collimation and horizontal axis, while 
the relevelling eliminated the error due to the error of adjust- 
ment of the plate-bubble. If this bubble were out of adjust- 
ment the vertical axis inclined as much to one side for the first 
setting as it did to the other side for the second setting. 

This operation may be repeated for a check, or to further 
eliminate errors of observation. The instrumental errors are 
wholly eliminated by one set of observations, as above given. 
It will be noted that this method is independent of the gradua- 
tion of the limb. The only assumptions are that the instru- 
ment and its adjustments are rigid during the reversal of the 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 97 

telescope, and that the pivots of the horizontal axis are true 
cylinders. 

ioi. Traversing. — A traverse, in surveying, is a series of 
consecutive courses whose lengths and bearings, or azimuths, 
have been determined. When a compass is used the bearing 
of each course is determined by the needle independently of 
that of the preceding course. When a transit is used and the 
needle not read, the graduated circle of the instrument is 
always oriented, or brought into the meridian, by taking a 
back-sight to the preceding station. If the azimuth* of the 
first course is known with reference to the meridian, the 
azimuth of all subsequent courses may be at once determined 
by properly orienting the limb of the instrument at the suc- 
cessive stations. Thus, if the south point has a zero azimuth 
the limb of the instrument should be oriented at each station, 
so that when the telescope points south vernier A shall read 
zero. 

The forward azimuth of a line is its angular deviation from 
the south point when measured at the rear station forward 
along the line. 

The back azimuth of a line is its angular deviation from the 
south point at the forward station when measured from that 
station back along the line. 

The forward and back azimuth differ by i8o° plus or minus 
the convergence of the meridians at the two extremities of the 
line. If this line is north and south it lies in the meridian, 
and hence its forward and back azimuth differ by 180 . When 
the course has an easterly or westerly component, or, in other 
words, when its extremities have different longitudes, the 
divergence of the line from the meridian at one end differs 
from its divergence from the meridian at the other by as much 

* In this treatise azimuth is always reckoned from the south point in the 
direction S.W.N. E. to 360 . The bearing of the line is thus given by its 
numerical value alone, without the aid of letters. 
7 



98 SURVEYING. 



as these meridians differ from parallelism. This is inappre- 
ciable on short lines, and hence in traversing the forward and 
back azimuth will be considered as differing by 180 . 

The field-work proceeds as follows, so far as the transit is 
concerned. Let it be assumed that from the initial point A 
of the survey the true azimuth to some other point Z is given. 
Let the stations be A, B, C, etc. 

Set vernier A to read the known azimuth AZ. With the 
alidade and limb clamped together, turn the telescope on Z 
and clamp the limb, setting carefully by means of the lower 
tangent-screw. If the alidade be now loosened and vernier A 
made to read zero, the telescope would point south. Turn 
the telescope on B by moving the alidade alone, and the read- 
ing of vernier A gives the forward azimuth of the line AB. 
Move the instrument to B and set vernier A to read the back 
azimuth of AB, which is found by adding 180^ to or subtract- 
ing it from the forward azimuth, according as this was less or 
more than 180 . With alidade and limb clamped at this read- 
ing, turn upon A, clamp the limb and unclamp the alidade, and 
the instrument is again properly oriented for reading directly 
the true azimuth of any line from this station, as the line BC, 
for instance. In this manner a traverse may be run with the 
transit, the field-notes showing the true azimuth of each course 
without reduction. The lengths of the courses may be found 
in any manner desired. 

If preferred, the telescope may be revolved on its horizon- 
tal axis and vernier A left with its forward reading, for orient- 
ing. Then revolve the telescope back to its normal position 
and proceed with the work.* 

* For a method of computing the coordinates of the courses, and the use of 
the traverse table, see chapter on Land Surveying. 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 99 



THE SOLAR ATTACHMENT. 

102. The Solar Attachment is a device to be fastened to 
the telescope axis of a transit-instrument, thus making a com- 
bination that will do the work of a solar compass. One form 
of this device is shown in Fig. 19.* The various spherical 
functions concerned in the problem are also represented in this 
figure by their several great circles. The polar axis, declination- 
arc, and collimation-arm are the same here as in the solar com- 
pass. The latitude-arc is here replaced by the vertical circle 
of the transit, and the telescope gives the line of sight. The 
adjustments and working of this attachment are so nearly iden- 
tical with those of the solar compass that they will not be 
repeated here. If the student has mastered the principles 
involved in the use of the solar compass he will have no diffi- 
culty in using the attachment. 

Various forms of solar attachments have been invented, the 
most recent and perhaps the most efficient of which is that 
shown in Fig. 20, invented by G. N. Saegmuller in 1881. It 
is manufactured by Fauth & Co., Washington, D. C, and by 
Keuffel & Esser, New York. It consists simply of an auxiliary 
telescope with bubble attached, having two motions at right 
angles to each other. These motions are horizontal and verti- 
cal when the main telescope, to which the attachment is rigidly 
fastened, is horizontal. If the main telescope be put in the 
meridian and elevated into the plane of the celestial equator, 
however, then the vertical axis of the attachment also lies in 
the meridian but points to the pole. It therefore becomes a 
polar axis about which the auxiliary telescope may revolve. If 
this telescopic line of sight be at right angles to the polar axis, 
it will generate an equatorial plane. If the line of sight be in- 
clined to this plane by an amount equal to and in the direction 
of the sun's declination, then when revolved on its polar axis it 

*From Gurley's Catalogue. 



100 



SUR VE YING. 




Fig. 19. 




Fig. 20. 



102 SURVEYING. 



would follow the sun's path in the heavens for the given day, 
provided the sun did not change its declination during the day. 
It only remains, therefore, to show how the latitude and decli- 
nation angles may be set off in order that the competency of 
this instrument to do the work of the solar compass may be- 
come apparent. 

To set off the declination-angle, turn the main telescope 
down or up according as the declination is north or south, and 
set the declination-angle on the vertical circle. Bring the 
small telescope into the plane of the large one and revolve it 
about its horizontal axis until its bubble comes to the centre 
of its tube. The angle formed by the two telescopic lines of 
sight is the declination-angle. Revolve the main telescope 
until it has an altitude equal to the co-latitude of the place, 
and clamp it in this position. With the vertical motions of 
both telescopes clamped, and their lateral motions free, if the 
line of sight of the small telescope can be brought upon the 
sun the main telescope must lie in the meridian. The vertical 
circle of the transit is thus seen to do the work of both the 
latitude and declination arcs of the solar compass. 

103. Adjustments of the Saegmuller Attachment. — 
First. All the adjustments of the transit must be as perfect 
as possible, but especially the plate and telescope bubbles, the 
the vernier of the vertical circle, and the transverse axis of the 
telescope. 

Second. To make the Polar Axis perpendicular to the Plane 
of the Line of Collimation and Horizontal Axis of the Main 
Telescope. — Carefully level the instrument and bring the teles- 
cope-bubble to the middle of its tube. The line of sight and 
horizontal axis of this telescope should now be horizontal, so 
that the polar axis is to be made vertical. To test this, revolve 
the auxiliary telescope about the polar axis, and see if the 
bubble on the small telescope maintains a constant position. 
If not, correct half the movement by means of the adjusting, 
screws at the base of the small disk, and the other half by re- 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 103 



volving the auxiliary telescope. These adjusting-screws are 
exactly analogous to the levelling-screws of the main instru- 
ment. 

Third. To make the Line of Sight of the Small Telescope 
parallel to the Axis of the Attached Bubble, — Make the large 
telescope horizontal by bringing its attached bubble to the 
middle of its tube. Bring the small telescope in the same plane 
and make it also horizontal by means of its bubble, clamping 
its vertical motion. Measure the vertical distance between the 
axis of the two telescopes, and lay off this distance on a piece of 
paper by two plain horizontal lines. Set this paper up at a con- 
venient distance from the instrument, and on about the same 
level. Bring the line of sight of the large telescope on the lower 
mark, and see if that of the small telescope falls on the upper 
mark. If not, adjust its reticule until its line of collimation 
does come on the upper mark. Revolve back to the horizontal 
to see if both bubbles again come to the middle simultaneously. 

When this adjustment is completed, there should be five 
lines in the instrument parallel to each other when instrument 
and telescopes are level, — viz., the axes of the two telescope- 
bubbles and of the plate-bubble on the standards, and the two 
lines of collimation, — and, in addition, the vernier on the vertical 
circle, should read zero. 

The seven adjustments (five of the transit and two of the 
attachment) must all be carefully made and frequently tested 
if the best results are desired. When this is done, this attach- 
ment will give the meridian to the nearest minute of arc, if ob- 
servations be taken when the sun is more than one hour from 
the horizon and two hours from the meridian. The advantages 
of the Saegmuller attachment consist mainly in having a teles- 
copic line of sight, and in the use of the vertical limb of the 
transit for setting off the declination and co-latitude. The 
effect of small errors in the latitude and declination angles, such 
as may be due to errors in the adjustments, is shown by the 
table, art. 54, p. 51. 



J 04 SURVEYING. 



THE GRADIENTER ATTACHMENT. 

104. The Gradienter is a tangent-screw with a micrometer- 
head attached to the horizontal axis of the telescope for the 
purpose of turning off vertical angles that are expressed in 
terms of its tangent as so many feet to the hundred. Such a 
device is shown in Fig. 17. In railroad work, the grade or slope 
is expressed in this manner, as 26.4 feet per mile, or as 0.5 foot 
per 100 feet. The micrometer-head is graduated so that one 
revolution raises or lowers the telescope by 1 foot or 0.5 foot 
in 100 feet. It is divided into 100 or 50 parts, so that each 
division on the head is equivalent to 0.01 foot in 100 feet. This 
attachment is found very convenient in railroad work. It is 
also of general utility in obtaining approximate distances. On 
level ground the distance is read directly, but on sloping 
ground the rod is still held vertical, and the distance read is too 
great. The true horizontal distance may be found by multiply- 
ing the distance read by the factors for horizontal distance 
given in table V.* Thus, if one revolution of the screw raises 
the line of sight 1 foot at a distance of 100 feet, and if at a cer- 
tain unknown distance one revolution of screw caused the line 
to pass over 5.5 feet on the rod, then the distance was 550 feet 
if the ground was horizontal. If the rod-readings had a mean 
vertical angle of 15 , the horizontal distance was 550 X 93.3 = 
513 feet. 

CARE OF THE TRANSIT. 

105. The Transit should be protected from rain and dust 
as much as possible. A silk gossamer water-proof bag should 
be carried by the observer to be used for this purpose. If water 
gets inside the telescope, remove the eye-piece and let it dry 
out. If moisture collects between the two parts of the objec- 



* This table is for reduction of stadia measurements, and is explained in 
the chapter on Topographical Surveying. 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 105 

tive remove it, and dry it with a gentle heat over a stove or 
lamp, but do not separate the glasses. If dust settles on the wires 
it may be blown off by removing both objective and eye-piece 
and blowing gently through the tube. Dust should be removed 
from the glasses by a camel's-hair brush, which should always 
be carried for the purpose. A clean handkerchief may be used 
with a gentle pressure to prevent scratching in case the dust is 
gritty. Use alcohol for cleansing greasy or badly soiled glasses. 
No part exposed to dust should be oiled, as this serves to retain 
all the dust that may fall on it. The centres should be cleaned 
occasionally with chamois skin, and oiled by a very little pure 
watch-oil. In the absence of watch-oil plumbago will be found 
to serve. A soft lead-pencil may be scraped and a little rubbed 
on the spindles with the finger. The tripod legs should have 
no lost motion either at the head or in their iron shoes. If the 
legs are split, as in Fig. 17, and fastened by thumb-nuts, these 
should be loosened when the instrument is carried and tight- 
ened again after setting. They may thus be made very tight 
and rigid, while the instrument is in use without danger of 
breaking the bolts in closing the legs, which is very liable to 
result if the screws are not loosened. For a method of putting 
in new cross-wires see chapter on Topographical Surveying. 

EXERCISES WITH THE TRANSIT. 

106. Establish three stations forming a triangle. Measure the three hori- 
zontal angles and see if their sum is 180 . 

107. Prolong a line in azimuth and distance by carrying both around an 
imaginary obstruction, and then check the azimuth by a back-sight and the dis- 
tance by measurement. Thus, let A and B be two points establishing a line. 
The problem is to establish two other points, Cand D, in the continuation of 
the line AB, with an imaginary obstruction to both sight and measurement 
between B and C. The distance BC is also to be obtained. 

The equilateral triangle will be found most efficient. 

108. Find both the distance to and the height of an inaccessible steeple, 
chimney, smokestack, or tree. 

Measure a base-line such that its two extremities make with the given object 



106 * SURVEYING. 



approximately an isosceles triangle (it is desirable that no angle of the triangle 

should be less than 30 nor more than 120°). The top of the object only need 

be visible from the two ends of the base. Measure both the horizontal and 

vertical angles at the extremities of the base-line subtended by the other two 

points of the triangle. Let A and B be the extremities of the base and P the 

point whose distance and elevation are required. We then have for horizontal 

angle? 

Sin P : sin A :: AB : BP\ 

also sin P : sin B :: AB : AP. 

In reading the vertical angles to the base-stations the reading should be 
taken on a point as high above the ground (or peg) as the telescope is above the 
peg over which it is set. The difference in the elevations of the two pegs is 
then obtained. The vertical angle to the point P is taken to the summit, and 
height of instrument added in each case to find its elevation above peg. If A 
be the lower of the two base-stations and if I a and Ib be the heights of instru- 
ment (line of sight) above the peg in the two cases, and if Va, Vb, Vp and 
Vp be the vertical angles read to the corresponding points, we may write: 

Elevation of B above A = AB tan Vb', 
" P " A—AP tan V P . 

Also, from the vertical angles taken at B, we have: 

Elevation of A below B = AB tan Va', 
" P above B = BP tan V P '. 

We now have a check on both the relative elevations and on the distances 
AP and BP. Assuming the elevation of A to be zero, we have: 

Elevation of P above A = AP tan Vp = AB tan Vb + BP tan V P '. 

This equality will not result unless the observations were well taken, the 
computations accurately made, and the instrument carefully adjusted. The ad- 
justments mainly involved here are the plate-bubbles and the vernier on the 
vertical circle. If the points are a considerable distance apart, as over a half- 
mile, the elevations obtained by reading the vertical angles are appreciably too 
great, on account of the earth's curvature. This may be taken as eight inches 
for one mile and proportional to the square of the distance. Or, we may write: 
Elevation correction on long sights, in inches,* = — 8 -^distance in miles. 
If the distances are all less than about half a mile, no attention need be paid 
to this correction in this problem. 



* For a full discussion of this subject see chap. XIV. 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. I07 

109. Find the height of a tree or house above the ground, on a distant hill, 
without going to the immediate locality. 

110. Find the horizontal length and bearing of a line joining two visible but 
inaccessible objects. Use the magnetic bearing if the true bearing of the base- 
line is not known. 

in. Find the horizontal length and bearing of a line joining two inaccessi- 
ble points both of which cannot be seen from any one position. 

Let A and B be the inaccessible points. Measure a base CD such that A is 
seen from C, and B from D. Auxiliary bases and triangles may be used to 
find the lengths of AC and BD. Knowing ^Cand CD and the included angle, 
compute AD in bearing and distance. The angle ADB may now be found, 
which, with the adjacent sides AD and BD known, enables the side AB to be 
found in bearing and distance. 

112. With the transit badly out of level, or with horizontal axis of the tele- 
scope thrown considerably out of the horizontal, measure the horizontal angle 
between two objects having very different angular elevations. Do this with 
both telescope normal and telescope reversed, and note the difference in the 
values of the angle obtained in the two cases. 

113. Select a series of points on uneven ground, enclosing an area, and 
occupy them successively with the transit, obtaining the traverse angles. That 
is, knowing or assuming the azimuth of the first line, obtain the azimuths of the 
other connecting lines, or courses, with reference to this one, returning to the 
first point and obtaining the azimuth of the first course as carried around by the 
traversed line. This should agree with the original azimuth of this course. 
The distances need not be measured for this check. 

114. Lay out a straight line on uneven ground by the method given in Art. 
ipo, occupying from six to ten stations. Return over the same line and estab- 
lish a second series of points, paying no attention to the first series, and then 
note the discrepancies on the several stakes. In returning, the two final points 
of the first line become the initial points of the second, this return line being a 
prolongation of the line joining these two points. If these deviate ever so 
little, therefore, from the true line, the discrepancy will increase towards the 
initial point. 

Similar exercises to those given for the solar compass may be assigned for 
the solar attachment. 



io8 



SURVEYING. 



THE SEXTANT. 

115. The Sextant is the most convenient and accurate 
hand-instrument yet devised for measuring angles, whether 
horizontal, vertical, or inclined. It is called a sextant because 
its limb includes but a 6o° arc of the circle. It will measure 
angles, however, to 120 . It is held in the hand, measures an 
angle by a single observation, and will give very accurate re- 
sults even when the observer has a very unstable support, as 




Fig. 21. 



on board ship. It is exclusively used in observations at sea, 
and is always used in surveying where angles are to be meas- 
ured from a boat, as in locating soundings, buoys, etc., as well 
as in reconnoissance work, explorations, and preliminary sur- 
veys. It has been in use since about 1730. 

The accompanying cut shows a common form of this in- 
strument as manufactured by Fauth & Co., Washington. The 
limb has a 7|-inch radius, and reads to 10 seconds of arc. 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. IO9 

There is a mirror M (Fig. 22), called the Index Glass, rigidly- 
attached to the movable arm MA, which carries a vernier 
reading on the graduated limb CD. There is another mirror, 
/, called the Horizon Glass, rigidly attached to the frame of 
the instrument, and a telescope pointing into this minor, also 
rigidly attached. This mirror is silvered on its lower half, but 
clear on its upper half. A ray of light coming from A passes 




Fig. 22. 



through the clear portion of the mirror /on through the tele- 
scope to the eye at E. Also, a ray from an object at O strikes 
the mirror M, is reflected to m, and then through the telescope 
to E. Through one half of the objective come the rays from 
H, and through the other half the rays from O, each of which 
sets of rays forms a perfect image. By moving the arm MA 
it is evident these images will appear to move over each other, 



110 SURVEYING. 



and for one position only will they appear to coincide. The 
bringing of the two images into exact coincidence is what the 
observation consists in, and however unsteady the motion of 
the observer may be, he can occasionally see both images at 
once, and so by a series of approximations he may finally put 
the arm in its true position for exact superposed images. 
The angle subtended by the two objects is then read off on 
the limb. 

116. The Theory of the Sextant rests on the optical 
principle that " if a ray of light suffers two successive reflec- 
tions in the same plane by two plane mirrors, the angle be- 
tween the first and last directions of the ray is twice the angle 
of the mirrors." 

To prove this, let OM and mE be the first and last posi- 
tions of the ray, the latter making with the former produced 
the angle E. The angle of the mirrors is the angle A. The 
angles of incidence and reflection at the two mirrors are the 
angles i and i\ PM, and pm being the normals. 

We may now write : 

Angle E = OMm — MmE. 

angle A = ImM — mMA 

== ( 9 o° - i') - ( 9 o° - i) 

Therefore E = 2A. Q. E. D. 

When the mirrors are brought into parallel planes, the 
angle A becomes zero, whence E also is zero, or the rays OM 
and Hm are parallel. This gives the position of the arm for 
the zero-reading of the vernier. The limb is graduated from 
this point towards the left in such a way that a 6o° arc of the 
circle will read to 120 . That is, a movement of i° on the arc 
really measures an angle of 2° in the incident rays, so it must 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS, III 

be graduated as two degrees instead of one. The very large 
radius enables this to be done without difficulty. 

ADJUSTMENTS OF THE SEXTANT. 

117. To make the Index Glass perpendicular to the 
Plane of the Sextant. — Bring the vernier to read about 30 
and examine the arc and its image in the index glass to see if 
they form a continuous curve. If the glass is not perpendi- 
cular to the plane of the arc, the image will appear above or 
below the arc, according as the mirror leans forward or back- 
ward. It is adjusted by slips of thin paper under the project- 
ing points and corners of the frame. 

118. To make the Horizon Glass Parallel to the Index 
Glass for a Zero-reading of the Vernier. — Set the vernier 
to read zero and see if the direct and reflected images of a 
well-defined distant object, as a star, come into exact coinci- 
dence. If not, adjust the horizon glass until they do. If this 
adjustment cannot be made, bring the objects into coincidence, 
or even with each other so far as the motion of the arm is con- 
cerned, and read the vernier. This is the index error of the 
instrument and is to be applied to all angles read. The better 
class of instruments all allow the horizon glass to be adjusted. 
This adjustment is generally given as two, but it is best con- 
sidered as one. If made parallel to the index glass after that 
has been adjusted, it must be perpendicular to the plane of 
the instrument. 

119. To make the Line of Sight of the Telescope 
parallel to the Plane of the Sextant. — The reticule in the 
sextant carries four wires forming a square in the centre of 
the field. The centre of this square is in the line of collima. 
tion of the instrument. 

Rest the sextant on a plane surface, pointing the telescope 
upon a well-defined point some twenty feet distant. Place two 
objects of equal height upon the extremities of the limb that 



112 SURVEYING. 



will serve to establish a line of sight parallel to the limb. Two 
lead-pencils of same diameter will serve, but they had best be 
of such height as to make this line of sight even with that of 
the telescope. It both lines of sight come upon the same 
point to within a half-inch or so at a distance of 20 feet, 
the resulting maximum error in the measurement of an angle 
will be only about 1". 

THE USE OF THE SEXTANT. 

120. To measure an Angle with the sextant, bring its 
plane into the plane of the two objects. Turn the direct line 
of sight upon the fainter object, which may require the instru- 
ment to be held face downwards, and bring the two images 
into coincidence. The reading of the limb is the angle re- 
quired. It must be remembered that the angles measured by 
the sextant are the true angles subtended by the two objects at 
the point of observation, and not the vertical or horizontal 
projection of these angles, as is the case with the transit. The 
true vertex of the measured angle is at E, Fig. 21. It is evident 
the position of E is dependent on the size of the angle, being 
at a great distance back of the instrument for a very small 
angle. The instrument should therefore not be used for meas- 
uring very small angles except as between objects a very great 
distance off. The sextant is seldom or never used for measur- 
ing angles where the position of the instrument (or the vertex 
of the angle) needs to be known with great accuracy. 

EXERCISES FOR THE SEXTANT. 

121. Measure the altitude of the sun or a star at its culmination by bringing 
the direct image, reflected from the surface of mercury held in a fiat dish on 
the ground, into coincidence with the image reflected from the index glass. 
Half the observed angle is the altitude of the body. The altitude of a terres- 
trial object may be obtained in the same manner, in which case the vessel of 
mercury should rest on an elevated stand ; the sextant could then be brought 
hear to it and the angular divergence of the two incident rays to the mercury 
surface and index glass reduced to an inappreciable quantity. 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. II3 

If the observation of a heavenly body be made on the meridian and the 
declination of the body at the time of observation be known, the latitude of the 
place is readily found. 

I2i#. Measure the angle subtended by two moving bodies, as of two men 
walking the street in the same direction, or of two boats on the water. (This is 
to illustrate the capacity of the sextant, for none but a reflecting instrument 
bringing two converging lines of sight into coincidence is competent to do 
this.) 

The exercises given in Arts. 106, 108, 109, and no for the transit may also 
serve for the sextant. Further applications of the sextant in locating soundings 
are given in chap. IX. 

122. The Double-reflecting Goniograph is a kind of dou- 
ble sextant and three-arm protractor* combined. It enables the 
two variable angles of the " three-point problem" f to be 
measured at once, and then provides for the immediate plot- 
ting of these angles upon the sheet, without reading off the 
values of the angles unless they are to be put on record. The 
angles may be read, however, and plotted afterwards if de- 
sired. This very ingenious and convenient instrument is the 
invention of Lieutenant Constantin Pott, of the English Navy. 
The construction and principles of the instrument are shown 
in Figs. 23, 24, and 25. To the graduated circle whose centre 
is D, Fig. 24, there are attached one fixed and two movable 
arms, each having one radial fiducial edge. The main frame- 
work of the instrument lies on the prolongation of the fixed 
arm A. Immediately back of the centre of the circle is a 
cylindrical frame containing two fixed mirrors, s s, one above 
the other, and also a free opening, W, Fig. 23. These corre- 
spond to the fixed mirror and clear glass on the sextant. Im- 
mediately back of these mirrors is the telescope, P, and on 
each side of this is a movable mirror, SS, attached to the slide 
bars //. These bars are fastened to the mirrors and slide 
freely through the studs Z Z^ set upon the movable arms B B r 

* For a description of the three arm protractor see chapter VI. 
f See chapter X., for a discussion of this problem. 
8 



114 



SURVEYING. 



The distance of these studs from the centre of the graduated 
circle is the same as that of the axes of the movable mirrors 
vS 6". Therefore a circle whose centre coincides with the centre 
of the graduated circle may pass through these four axes. 




Figs. 23 and 24. 



The theory of the instrument is shown in Fig. 25. The ray 
of light R is reflected from c to e and thence down the tele- 
scope to A. The object in the prolongation of AB casts the 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 1 1 5 



ray Be directly down the telescope. The angle formed by 
the incident and final reflected ray, Rfe, is twice the angle 
subtended by the plains of the mirrors C g e y as was shown in 
the case of the sextant. When the rays R and B coincide the 
mirrors 5 S and s s, Fig. 24, are parallel. The slide-bar then 
has the position Ca. When the arm has come into the posi- 
tion dB', making the angle with the fixed arm dB, the slide- 




Fig. 25. 

bar has come into the position Ca', making an angle \(J> with 
its former position ca, since this is an angle in the circumfer- 
ence. The mirror SS has also turned through an angle |-0 
and since it was parallel to the mirror ss in its first position it 
now makes an angle y = \ <p with it. The angle /3, which is 
the angle subtended by the incident ray Re and by the direct 
ray BA, is therefore equal to the angle 0, which is the angle 
ada' read on the graduated circle. 



u6 



SUR VE YING. 



Both movable arms are provided with clamp-screws, K K, 
and tangent screws, M M. The instrument is held, while 
observing, by the handle H, Fig. 23 ; but when used for plot- 
ting the point of observation this handle is unshipped and the 
instrument manipulated by the two milled heads F and G. 
The centre at d, Fig. 24, is open, so that when the instrument 
is adjusted to the plotted positions of the three known stations, 
the point of observation is marked by a pencil through the 
open centre. It is therefore a double sextant for observing 
and a three-arm protractor for plotting. 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 117 



CHAPTER V. 

THE PLANE TABLE. 

123. The Plane Table consists of a drawing-board 
properly mounted on which rests an alidade carrying a line of 
sight rigidly attached to a plain ruler with a fiducial edge. 
The line of sight is usually determined by a telescope, as in 
Fig. 26. This telescope has no lateral motion with respect to 
the ruler, but both may be moved at pleasure on the table. 
The telescope has a vertical motion on a transverse axis, as in 
the transit. It is also provided with a level tube, either 
detachable or permanently fixed. The table is levelled by 
means of one round or two cross bubbles on the ruler of the 
alidade. The line of sight of the telescope is usually parallel 
to the fiducial edge of the ruler, though this is not essential. 
It is only necessary that they should have a fixed horizontal 
angle with each other. The table itself must have a free hori- 
zontal angular movement and the ordinary clamp and slow- 
motion screw. The table corresponds to the graduated limb 
in the transit, the alidades in the two instruments performing 
similar duties. Instead, however, of reading off certain hori- 
zontal angles, as is done with the transit, and afterwards 
plotting them on paper, the directions of the various pointings 
are at once drawn on the paper which is mounted on the top 
of the table, no angles being read. The true relative positions 
of certain points in the landscape are thus transferred directly 
to the drawing-paper to any desired scale. The magnetic 
bearing of any line may be determined by means of the decli- 
nator, which is a small box carrying a needle which can swing 
some ten degrees either side of the zero-line. The zero-line 



u8 



SURVEYING. 







Fig. 26. 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. II9 

being parallel to one edge of the box, the magnetic meridian 
may be at once marked down on any portion of the map, and 
the bearing of any intersecting line determined by means of a 
protractor. The instrument has been long and extensively used 
for mapping purposes, and is still the only instrument used 
for the " filling-in" of the topographical charts of the U. S. 
Coast and Geodetic Survey. An extended account of the 
instrument and the field methods in use on that service may 
be found in Appendix 13 of the Report of the U. S. Coast and 
Geodetic Survey for 1880. The following discussion is partly 
from that source. 

ADJUSTMENTS OF THE ALIDADE. 

124. To make the Axes of the Plate-bubbles parallel 
to the Plane of the Table. — Level the table with the alidade 
in any position, noting the readings of the bubbles. Mark the 
exact position of the alidade on the table, take it up carefully, 
and, reversing it end for end, replace it by the same marks. If 
the bubbles now have the same readings as before, with refer- 
ence to the table they are parallel to the plane of the table. 
If not, adjust the bubbles for one half the movement and try 
again. 

125. To cause the Line of Sight to revolve in a Vertical 
Plane. — This adjustment is the same as in the transit. It need 
not be made with such extreme accuracy, however, and the 
plumb-line test is sufficient. With the instrument carefully 
levelled, cause the line of sight to follow a plumb-line through 
as great an arc as convenient. If the line of sight deviates 
from the plumb-line raise or lower one end of the transverse 
axis of the telescope, until it will follow it with sufficient exact- 
ness. 

126. To cause the Telescope-bubble and the Vernier on 
the Vertical Arc to read Zero when the Line of Sight is 
Horizontal. — This adjustment is also the same as in the 



120 SURVEYING. 



transit. The methods given for the transit may be used with 
the plane table, or a sea horizon may be used as establishing a 
horizontal line, or a levelling-instrument may be set up beside 
the plane table having the telescopes at the same elevation, and 
both lines of sight turned upon the same point in the horizontal 
plane as determined by the level. The bubble and vernier are 
then both adjusted to this position of telescope. 

This adjustment is important if elevations are to be deter- 
mined either by vertical angles or by horizontal lines of sight. 
If only geographical position is sought this adjustment may 
be neglected. 

THE USE OF THE PLANE TABLE. 

127. In using the Plane Table at least two points on the 
ground, over which the table may be set, must be plotted on 
the paper to the scale of the map before the work of locating 
other points can begin. This requires that the distance between 
these points shall be known, which distance becomes the base- 
line for all locations on that sheet. Any error in the measure- 
ment or plotting of this line produces a like proportional error 
in all other lines on the map. 

The plane table is set over one of these plotted points, the 
fiducial edge of the ruler brought into coincidence with the two 
points, and the table revolved until the line of sight comes on 
the distant point. The table is now clamped and carefully set 
by the slow-motion screw in this position, when it is said to be 
oriented, or in position. 

In Figs. 27 and 28, let T, T/ T," T, m represent the plane- 
table sheet and the points a and/ the original plotted points. 
The corresponding points on the ground are A and P, the latter 
being covered by/ in Fig. 27, and the former by a in Fig. 28. 
In Fig. 27, the plotted point/ is centred over the point P, the 
ruler made to coincide with ap, and the telescope made to read 
on A by shifting the table. For plotting the directions of 






121 




1, 



Fig. 27. 



h 




F10. s8. 



122 SURVEYING. 



other objects on the ground, the alidade is made to revolve about 
p just as the transit revolves about its centre. A needle is 
sometimes stuck at this point, and the ruler caused to press 
against it in all pointings, but this defaces the sheet. Other 
pointings are now made to B, C, and D, which may be used as 
stations, and also to a chimney (c/i.), a tree (/.), a cupola 
{cup.), a spire (sp.), and a windmill (w.m.). Short lines are 
drawn at the estimated distance from/, and these marked with 
letters, as in the figure, or by numbers, and a key to the numbers 
kept in the sketch- or note-book. 

The table is now removed to A, the other known point, and 
set with the point a on the plot over the point A on the 
ground, when the table is approximately oriented. The ruler 
is now set as shown in Fig. 28, coinciding with a and /, but 
pointing towards/. The table is then swung in azimuth until 
the line of sight falls on P, when it is clamped. It is now 
oriented * for this station, and pointings are taken on all the 
objects sighted from P y and on such others as may be sighted 
from subsequent stations, the alidade now revolving about the 
point a on the paper. The intersections on the plot of the 
two pointings taken to the same object from A and P will evi- 
dently be the true position on the plot for those points with 
reference to, and to the scale of, the line ap. These intersec- 
tions are shown in Fig. 28. 

It is evident that if other points, as D or C, be now occu- 
pied, the table oriented on either A or P, and pointings taken 
on any of the objects sighted from both A and P, the third or 
fourth line drawn to the several objects should intersect the 
first two in a common point. This furnishes a check on the 
work, and should be taken for all important points. It is pref- 
erable also to have more than two points on the sheet pre- 

* It will be noted that this process of orienting the plane table is practically 
identical with that by which the limb of the transit is oriented in traversing 
(art. 101). 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 123 

viously determined. Thus, if B were also known and plotted 
at b, when the table had been oriented on any other station, 
and a pointing taken to B, the fiducial edge of the ruler 
should have passed through b. 

As fast as intersections are obtained and points located 
the accompanying details should be drawn in on the map to 
the proper scale. If distances are read by means of stadia 
wires on a rod held at the various points (see chap. VIII), 
then a single pointing may locate an object, the distance being 
taken off from a scale of equal parts, and the point at once 
plotted on the proper direction-line. It is now common to do 
this in all plane-table surveying. 

~* 128. Location by Resection. — This consists in locating 
the points occupied by pointings to known and plotted points. 
The simple case is where a single pointing has been taken to 
this point from some known point, and a line drawn through 
it on the sheet. It is not known what point on this line 
represents the plotted position of this station. The setting of 
the instrument can therefore be but approximate, but near 
enough for all purposes. The table can be oriented as before, 
there being one pointing and corresponding line from a known 
point. A station is then selected, a pointing to which is as 
nearly 90 degrees from the orienting line as possible, and the 
alidade so placed that while the telescope sights the object the 
fiducial edge of the ruler passes through the plot of the same 
on the sheet. The intersection of this edge with the former 
line to this station gives the station's true position on the 
sheet. This latter operation is called resection. Another re- 
section from any other determined point may be made for a 
check. 

129. To find the Position of an Unknown Point by Re- 
section on Three Known Points. — This is known as the 
Three-point Problem, and occurs also in the use of the sextant 
in locating soundings. It is fully discussed in that connection 



124 SURVEYING. 



(see chap. X.), so that only a mechanical solution suitable 
for the problem in hand will be given here. It is under- 
stood there are three known points, A, B, and C, plotted in 
a, b, and c on the map. The table is set up over any given 
point (not in the circumference of a circle through A, B, and 
C). Fasten a piece of tracing-paper, or linen, on the board, 
and mark on it a point / for the station P occupied. Level 
the table, but of course it cannot be oriented. Take pointings 
to A, B, and C, and draw lines on the tracing-paper from p 
towards a, b, and c y long enough to cover these distances when 
drawm to scale. Remove the alidade and shift the tracing- 
paper until the three lines drawn may be made to coincide 
exactly with the three plotted points a, b, and c. The point 
p is then the true position of this point on the sheet. This 
being pricked through, the table may now be oriented and the 
work proceed as usual. 

130. To find the Position of an Unknown Point by Re- 
section on Two Known Points. — This is called the Two- 
point Problem, and but one of several solutions will be given. 
It is evident that if the table could be properly oriented over 
the required point, its position on the sheet could be at once 
found by resection on the two known points. The table may 
be oriented in the following manner : Let A and B be the 
known points plotted in a and b on the sheet. Let C be the 
unknown point whose position c on the sheet is desired. 
Select a fourth point D, which may be occupied, and so placed 
that intersections from C and D on A and B will give angles 
between 30 and 120 degrees. Fasten a piece of tracing linen 
or paper on the board, marking a point d' at random. Set 
up over D, orienting the table as nearly as may be by the 
needle or otherwise. Draw lines from d' towards A, B, and 
C. Mark off on the latter the estimated distance to C, to 
scale, calling this point ' . Set up over C, with c r over the 
station, orienting on D by the line c'd' . This brings the table 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 12$ 

parallel to its former position at D. From c' draw lines to A 
and B, intersecting the corresponding lines drawn from d' in 
a' and b'. We now have a quadrilateral a'b'c'd' similar to 
the quadrilateral formed by the true positions of the plotted 
points abed, but it differs in size, since the distance c'd' was 
assumed, and also in position (azimuth), since the table was 
not properly oriented at either station. Remove the alidade, 
and shift the tracing until the line a'b' coincides with a and b 
on the sheet. Replace the alidade on the tracing, bringing it 
into coincidence with c'a! , e'b', or c'd' , and revolve the table on 
its axis until the line of sight comes upon A, B, or D, as the 
case may be. The table is now oriented, when the true posi- 
tion of c may be readily found by resecting from a and b } 
which, when pricked through, gives its position on the sheet. 

The student may show how the same result could have been obtained with- 
out the aid of tracing-paper. 

If the fourth point D may be taken in range 'with A and B, 
the. table may be properly oriented on this range, and a line 
drawn towards C from any point on this range line on the plot. 
Then C is occupied, and the table again properly oriented by 
this line just drawn, when the true position of c may be found 
by resecting from a and b, as before. 

In general, if the table can be properly oriented over any 
unknown point from which sights may be taken to two or 
more known (plotted) points, the position of this unknown 
point is at once found by resection from the known points. 

The student would do well to look upon the table and the 
attached plot as analogous to the graduated horizontal limb in 
the transit. The principles and methods of orienting are pre- 
cisely similar, the pointings differing only in this, that with the 
transit the horizontal angle, referred to the meridian, is read 
off, recorded, and afterwards plotted, while with the plane 
table this bearing is immediately drawn upon the sheet. 

131. The Measurement of Distances by Stadia. — This 



126 SURVEYING. 



method of determining short distances is now generally used in 
connection with the plane table. It is fully discussed in chap- 
ter VIII., where the principles of its action and its use with 
the transit are given at length. The same principles, field 
methods, and tables apply to its use with the plane table, 
with such modifications as one accustomed to the use of the 
plane table would readily introduce. When used in this way 
it enables a point to be plotted from a single pointing, it 
being located by polar coordinates (azimuth and distance), in- 
stead of by intersections. 

EXERCISES WITH THE PLANE TABLE. 

132. Make a plane table survey of the college campus, measuring the length 
of one side for a base. 

133. Having located several points on the sheet by intersections, occupy 
them and check their location by resection. 

134. Locate a point (not plotted) by resection on three known points (art. 
129). 

135. Locate a point (not plotted) by resection on two known points, first 
taking the auxiliary point D not in line with AB, and then by taking it in line 
with AB. This gives a check on the position of the point, and shows the ad- 
vantages of the second method when it is feasible. 






ADJUSTMENT, USE, AND CARE OF INSTRUMENTS'. 12J 



CHAPTER VI. 

ADDITIONAL INSTRUMENTS USED IN SURVEYING AND 

PLOTTING. 

THE ANEROID BAROMETER. 

136. The Aneroid Barometer consists of a circular me- 
tallic box, hermetically sealed, one side being covered by a 
corrugated plate. The air is mostly removed, enough only 
being left in to compensate the diminished stiffness of the cor- 




Fig. 29. 

rugated cover at higher temperatures. This cover rises or 
falls as the outer pressure is less or greater, and this slight 
motion is greatly multiplied and transmitted to an index 
pointer moving over a scale on the outer face. The motion 
of the index is compared with a standard mercurial thermom- 
eter and the scale graduated accordingly. Inasmuch as all 



128 SURVEYING. 



barometric tables are prepared for mercurial barometers, 
wherein the atmospheric pressure is recorded in inches of 
mercury, the aneroid barometer is graduated so that its read- 
ings are identical with those of the mercurial column. 

Figure 29 shows a form of the aneroid designed for eleva- 
tions to 4000 feet above or to 2000 feet below sea-level. It 
has a vernier attachment and is read with a magnifying-glass 
to single feet of elevation. It must not be supposed, how- 
ever, that elevations can be determined with anything like this 
degree of accuracy by any kind of barometer. The barometer 
simply indicates the pressure at the given time and place, but 
for the same place the pressure varies greatly from various 
causes. All barometric changes, therefore, cannot be attrib- 
uted to a change in elevation, when the barometer is carried 
about from place to place. 

If two barometers are used simultaneously, which have 
been duly compared with each other, one at a fixed point of 
known elevation and the other carried about from point to 
point in the same locality, as on a reconnoissance, then the 
two sets of readings will give very close approximations to 
the differences of elevation. If the difference of elevation be- 
tween distant points is desired, then long series of readings 
should be taken to eliminate local changes of pressure. The 
aneroid barometer is better adapted to surveys than the mer- 
curial, since it may be transported and handled with greater 
ease and less danger. It is not so absolute a test of pressure, 
however, and is only used by exploring or reconnoissance 
parties. For fixed stations the mercurial barometer is to be 
preferred. It has been found from experience that the small 
aneroids of if to 2\ inches diameter give as accurate results 
as the larges ones. 

137. Barometric Formulae. — In the following derivation 
of the fundamental barometric formula, the calculus is used so 
that the student will have to take portions of it on trust until 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 1 29 

he has studied that branch of mathematics. All that follows 
Eq. (4) he can read. 

Let H= height of the " homogeneous atmosphere"* in lat. 

45°- 
h = corresponding height of the mercurial column. 

3 = the relative density of the " homogeneous atmos- 
phere" with reference to mercury. 
z = difference of elevation between two points, with 
barometric readings of h' and h v at the higher 
and at the lower point respectively. 
Then from the equilibrium between the pressures of the 
mercurial column and atmosphere we have : 

h = 6H . . . (1) 

Also, for a small change in elevation, dz, the corresponding 
change in the height of the mercurial column would be 

dh = $dz (2) 

Substituting in (2) the value of d as given by (1), we have : 

dh = -jydz ; 

£2 

or, dg = H T (3) 

Integrating (3) between the limits h' and h x we have: 

2 = H £ ^T = H 1o S«J> • • • ' (4) 

* " Homogeneous atmosphere" signifies a purely imaginary condition 
wherein the atmosphere is supposed to be of uniform density from sea-level to 
such upper limit as may be necessary to give the observed pressure at the ob- 
served temperature. 
9 



ISO SURVEYING. 



where the logarithm is in the Napierian system. Dividing by 
the modulus of the common system to adapt it to computation 
by the ordinary tables, we have : 

h 

2= 2.30258^ log a j± (5) 

If H be the height of the homogeneous atmosphere at a 
temperature of 32 F., and if k be the height of the mercurial 
column at sea-level at same temperature, and if g m and g a be 
the specific gravities of mercury and air respectively, then, 
evidently, 

or, H =^- (6) 

From experiment we have : 

k = 29.92 inches, 

£•»= I3-596 

g a — O.OOI239 " 

whence H = 26,220 feet. 

This is on the assumption that gravity is constant to this 
height above sea-level. When this is corrected for variable 
gravity we have : 

H = 26,284 feet (7) 

Equati6n (7) gives the height of the homogeneous atmos- 
phere at a temperature of 32 F. But since the volume of a gas 
under constant pressure varies directly as the temperature, and 
since the coefficient of expansion of air is 0.002034 for i° F., 
we have for the height of the homogeneous atmosphere at any 
temperature : 

H=H [1 + 0.002034 (/ - 3 2 )] ... (8) 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. I3I 

If the temperature chosen be the mean of the temperatures at 
the two points of observation, as t x and t t for the upper and 
lower points respectively, then we should have : 

H=H fi + 0.002034 (-y--32J 

= 26,284 [1 + 0.001017 (*'+ t x — 64)] . . (9) 

Substituting this value of H in Eq. (5) we obtain : 

h 

# = 60,520 [I +0.00I0I7 {f'+t x — 64)] l0gyf. . (IO) 

If we wish to refer this equation to approximate sea-level 
(height of mercurial column of 30 inches) and to a mean tem- 
perature of the two stations of 50 F., we may write : 

3? 

1 K 1 h' 1 3° , 30 

l°g£,=log-=log T/ -log T; 

Also, when t' -\-t 1 = ioo°, we have 

t + 1, - 64 = 36 . 
Substituting these equivalents in eq. (10), we obtain 

# = 60520(1 +0.001017X36) (log ~ - log j), 



# = 62737 ^g ^ - 62737 log j (II) 



132 SURVEYING. 






In this equation, the two terms of the second member rep- 
resent the elevations of the upper and lower points respec- 
tively, above a plane corresponding to a barometric pressure of 
30 inches and for a mean temperature of the two positions of 
50° F. 

Table I. is computed from this equation, the arguments be- 
ing the readings of the barometer, h' and h v at the upper and 
lower stations respectively, the tabular results being elevations 
above an approximate sea-level. The difference between the 
two tabular results gives the difference of elevation of the two 
points, for a mean temperature of 50 and no allowance made 
for the amount of aqueous vapor in the air. For other tem- 
peratures, and for the effect of the humidity (which is not ob- 
served,, but the average conditions assumed to exist), a certain 
correction needs to be applied, which correction is not an abso- 
lute amount,, but is always a certain proportion of the difference 
of elevation as obtained from eq. (11) or table I. If the two 
elevations taken from the table be called A' and A v and the 
correction for temperature and humidity be C, we would have 

z=(A'-A 1 ){i + C). ..... (12) 

It is seen, therefore, that C is a coefficient which, when mul- 
tiplied into the result obtained from table I., gives the correc- 
tion to be applied to that result. The values of C are given 
in table II. for various values of t' -f- t v 

The following example will illustrate the use of the tables : 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 1 33 



TABLE I. BAROMETRIC ELEVATION.* 

3° 
Containing A = 62737 l°g ~r • Argument, h. 



h. 


A. 


Dif. for 
.01. 


k. 


A. 


Dif. for 

.CI. 


h. 


A. 


Dif. for 
.01. 


Inches. 


Feet. 


Feet. 


Inches. 


Feet. 


Feet. 


Inches. 


Feet. 


Feet. 


II. O 


•2-1^ 


—24.6 


I4.O 


20,765 


-19-5 


17- 





15,476 


-I6.0 


II. I 


27,090 


24.4 


14. 1 


20,570 


19-3 


17. 


1 


15,316 


15 


9 


II. 2 


20,846 


24.2 


14.2 


20,377 


19. 1 


17. 


2 


15,157 


15 


8 


11. 3 


26,604 


24.O 


14.3 


20,186 


18.9 


17- 


3 


J 4,999 


15 


7 


11. 4 


26,364 


23.8 


I4.4' 


19,997 


18.8 


17. 


4 


14,842 


15 


6 


"•5 


26,126 


23.6 


14.5 


19,809 


18.6 


17- 


5 


14,686 


15 


5 


11. 6 


25,890 


23-4 


I4.6 


19.623 


18.6 


17. 


6 


I4,53i 


15 


4 


11. 7 


25,656 


23.2 


I4.7 


19,437 


18.5 


17- 


7 


14,377 


15 


4 


11. 8 


25,424 


23.0 


I4.8 


19,252 


18.4 


17 


8 


14,223 


15 


3 


11. 9 


25,194 


22.8 


I4.9 


19,068 


18.2 


17 


9 


14,070 


15 


2 


12.0 


24,966 


22.6 


I5.0 


18,886 


18. 1 


18 





13,918 


15 


1 


12. 1 


24,740 


22.4 


i5-i 


18,705 


18.0 


18 


1 


13.767 


15 





12.2 


24,516 


22.2 


15-2 


18,525 


17.9 


18 


2 


13,617 


14 


9 


12.3 


24,294 


22.1 


15.3 


18,346 


17.8 


18 


3 


13,468 


14 


9 


12.4 


24,073 


21.9 


15.4 


18,168 


17.6 


18 


4 


13,319 


14 


7 


12.5 


23,854 


21.7 


15.5 


17,992 


17-5 


18 


5 


13,172 


14 


7 


12.6 


23,637 


21.6 


15.6 


17,817 


17.4 


18 


6 


13,025 


14 


6 


12.7 


23,421 


21.4 


15.7 


17,643 


17.3 


18 


7 


12,879 


14 


6 


12.8 


23,207 


21.2 


15.8 


17,470 


17.2 


18 


8 


12,733 


14 


4 


12.9 


22,995 


21.0 


15-9 


17,298 


17. 1 


18 


9 


12,589 


14 


4 


13-0 


22,785 


20.9 


16.0 


17,127 


16.9 


r 9 





12,445 


14 


3 


I3-I 


22,576 


20.8 


16. 1 


16,958 


16.9 


*9 


1 


12,302 


14 


2 


13-2 


22,368 


20.6 


16.2 


16.789 


16.8 


19 


2 


12,160 


14 


2 


13-3 


22,l62 


20.4 


16.3 


l6,62I 


16.7 


19 


3 


12.018 


14 


1 


13.4 


21,958 


20.1 


16.4 


16.454 


16.6 


19 


4 


11,877 


14 





13.5 


21,757 


20.0 


16.5 


16,288 


16.4 


19 


5 


ii,737 


13 


9 


13.6 


21,557 


19.9 


16.6 


16,124 


16.3 


19 


.6 


n,598 


13 


9 


13.7 


21,358 


19.8 


16.7 


15 96l 


16.3 


19 


•7 


ii,459 


13 


.8 


13.8 


21,160 


19.8 


16.8 


15.798 


T6.2 


19 


.8 


11,321 


13 


•7 


13.9 


20,962 


-19.7 


16.9 


15,636 


-16.0 


19 


■9 


11,184 


-13 


7 


14.0 


20.765 




17.0 


15.476 




20 


.0 


11.047 





* This table 
Survey, 1881. 



taken from Appendix 10, Report U. S. Coast and Geodetic 



134 



SURVEYING. 



TABLE I. Barometric Elevation. — Continued. 





Containing A = I 


32737 log 


3° 

— .. Argument, h. 






k. 


A. 


Dif. for 

.01. 


h. 


A. 


Dif. for 
.01. 


h. 


A. 


Dif. for 

.01. 


Inches. 


Feet. 


Feet. 


Inches. 


Feet. 


Feet. 


Inches. 


Feet. 


Feet. 


20. 


11,047 


-13-6 


23.O 


7,239 


-11. 8 


26.O 


3,899 


— IO.5 


20.I 


10,911 


13-5 


23.1 


7,121 


11. 7 


26.1 


3,794 


IO.4 


20.2 


10,776 


13.4 


23.2 


7,004 


11. 7 


26.2 


3,690 


IO.4 


20.3 


10,642 


13-4 


23.3 


6,887 


• 

11. 7 


26.3 


3,586 


IO.3 


20.4 


IO,508 


13-3 


23-4 


6,770 


'11. 6 


26.4 


3,483 


IO.3 


20.5 


IO,375 


13.3 


23.5 


6,654 


11. 6 


26.5 


3,38o 


IO.3 


20.6 


10,242 


13.2 


23.6 


6,538 


n-5 


26.6 


3,277 


10.2 


20.7 


IO,IIO 


I3-I 


23-7 


6,423 


11. 5 


26.7 


3,175 


I0.2 


20.8 


9.979 


13. 1 


23.8 


6,308 


11. 4 


26.8 


3,o73 


IO. I 


20.9 


9,848 


13.0 


23-9 


6,194 


11. 4 


26.9 


2,972 


IO. I 


21.0 


9,718 


12.9 


24.0 


6,080 


11. 3 


27.O 


2,871 


IO. I 


21. 1 


9.589 


12.9 


24.1 


5,967 


n-3 


27.I 


2,770 


IO. O 


21.2 


9,460 


12.8 


24.2 


5,854 


11. 3 


27.2 


2,670 


IO. O 


21.3 


9332 


12.8 


24.3 


5,741 


11. 2 


27-3 


2.570 


IO. O 


21.4 


9.204 


12.7 


24.4 


5,629 


11. 1 


27.4 


2,470 


9.9 


21-5 


9,°77 


12.6 


24-5 


5,518 


11. 1 


27-5 


2,37i 


9.9 


21.6 


8,95i 


12.6 


24.6 


5,407 


11. 1 


27.6 


2,272 


9.9 


21.7 


8,825 


12.5 


24.7 


5,296 


11. 


27.7 


2,173 


9.8 


21.8 


8,700 


12.5 


24.8 


5,186 


10.9 


27.8 


2,075 


9.8 


21.9 


8,575 


12.4 


24.9 


5.077 


10.9 


27.9 


i,977 


9-7 


22.0 


8,45i 


12.4 


25.0 


4,968 


10.9 


28.0 


1,880 


9-7 


22.1 


8,327 


12.3 


25.1 


4,859 


10.8 


28.1 


1,783 


9-7 


22.2 


8,204 


12.2 


25.2 


4,751 


10.8 


28.2 


1,686 


9-7 


22.3 


8,082 


12.2 


25-3 


4,643 


10.8 


28.3 


1,589 


9.6 


22.4 


7,960 


12.2 


25.4 


4,535 


10.7 


28.4 


1,493 


9.6 


22.5 


7,838 


12. I 


25.5 


4,428 


10.7 


28.5 


1,397 


9-5 


22.6 


7,717 


I2.0 


25.6 


4,32i 


10.6 


28.6 


1,302 


9-5 


22.7 


7,597 


I2.0 


25-7 


4,215 


10.6 


28.7 


1,207 


9-5 


22.8 


7,477 


II. 9 


25.8 


4,109 


10.5 


28.8 


1,112 


9.4 


22.9 


7,358 


-11. 9 


25.9 


4,004 


-10.5 


28.9 


1,018 


-9.4 


23.O 


7-239 




26.0 


3,899 




29.0 


924 





ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 1 35 



TABLE I. Barometric Elevations. — Continued. 

30 

Containing A == 62737 l°g "T • Argument, h. 



h. 


A. 


Dif. for 


h. 


A. 


Dif. for 


h. 


A. 


Dif. for 






.01. 






.01. 






.01. 


Inches. 


Feet. 


Feet. 


Inches. 


Feet. 


Feet. 


Inches. 


Feet. 


Feet. 


29.O 


924 


-9.4 


29.7 


274 


-9.2 


30.4 


361 


—9.0 


29.I 


830 


9.4 


29.8 


182 


9.1 


30.5 


451 


8.9 


29.2 


736 


9-3 


29.9 


91 


9.I 


30.6 


540 


8.9 


29-3 


643 


9-3 


30.0 


OO 


9.1 


30.7 


629 


8.8 


29.4 


550 


9.2 


30.1 


-91 


9.0 


30.8 


717 


8.8 


29-5 


458 


9.2 


30.2 


l8l 


9.0 


30.9 


805 


-8.8 


29.6 


366 


-9.2 


30.3 


271 


-9.0 


3I.O 


-893 




29-7 


274 




30-4 


361 











TABLE II. 

CORRECTION COEFFICIENTS TO BAROMETRIC ELEVATIONS 
FOR TEMPERATURE AND HUMIDITY.* 



h + 1'. 


c. 


h + 1' : 


c. 


h + 1>. 


c. 


o° 


—0. 1025 


60 


—0.0380 


120 


-f-0.0262 


5 


— .0970 


65 


— .0326 


125 


+ .0315 


10 


— -0915 


70 


— .0273 


130 


+ .0368 


15 


— .0860 


75 


— .0220 


135 


-j- .0420 


20 


— .0806 


80 


— .0166 


140 


+ -°472 


25 


- -0752 


85 


— .0112 


145 


+ -°524 


30 


— .0698 


90 


— .0058 


150 


+ -0575 


35 


- .0645 


95 


— . 0004 


155 


-j- .0626 


40 


- .0592 


100 


-f- .0049 


160 


+ - o6 77 


45 


- -0539 


105 


-]- .0102 


165 


-\- .0728 


5o 


— .0486 


• r o 


+ .0156 


170 


+ -°779 


55 


- .0433 


115 


+ .0209 


175 


-\- .0829 


60 


— .0380 


120 


+ .0262 


180 


+ .0879 



*This table compiled from tables I. and IV. of Appendix 10 of Report of 
the U. S. Coast and Geodetic Survey for 1881. 



1 36 SUR VE YING. 



Example, 

From observations made at Sacramento, CaL, and at Sum- 
mit on the top of the Sierra Nevada Mountains, the annual 
means were : 



h! 


= 23. 


288 in. 




t' 


= 42.1 


F. 


K 


= 30. 


014 in. 




*i 


= 59-9- 




I. 


we h; 


ave 














A' = 


69 


01.0 


feet. 








A x = 


— 


12.7 


a 





A'— A, = 6913.7 " 

From table II. we find for t' -\- t x = i02°.o, C= + .0070. 
. • . z = 6913.7 (1 -|- .0070) == 6962 feet. 

138. Use of the Aneroid. — The aneroid barometer should 
be carried in a leather case, and it should not be removed from 
it. It should be protected from sudden changes of tempera- 
ture, and when observations are made it should have the 
temperature of the surrounding outer air It should not be 
carried so as to be affected by the heat of the body, and should 
be read out of doors, or at least away from all artificially 
warmed rooms. Always read it in a horizontal position. The 
index should be adjusted by means of a screw at its back, to 
agree with a standard mercurial barometer, and then this ad- 
justment left untouched. 

When but a single instrument is used it is advisable to pass 
between stations as rapidly as possible, but to stop at a number 
of stations during the day for a half-hour or so, reading the 
barometer on arrival and on leaving. The difference of these 
two readings shows the rate of change of barometric readings 
due to changing atmospheric conditions, and from these iso- 
lated rates of change a continuous correction-curve can be con- 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 1 37 

structed on profile or cross-section paper from which the 
instrumental corrections can be taken for any hour of the 
day.* The observations should be repeated the same day in 
reverse order, the corrections applied as obtained from this 
correction curve, a'nd the means taken. Observations should 
be made when the humidity of the air is as nearly constant as 
possible, and never in times of changeable or snowy weather. 
Let the student measure the heights of .buildings, hills, etc., 
and then test his results by level or transit. 

THE PEDOMETER. 

139. The Pedometer is a pocket-instrument for register- 
ing the number of paces taken when walking. It is generally 





Fig. 30. — Front View. 



Fig. 31.— Back View. 



made in the form of a watch, the front and back views being 
shown in Figs. 30 and 31. 



* Mr. Chas. A. Ashburner, Geologist of the Penn. Geol. Survey, has used 
this method with good results. 



138 



SUI? VE YING. 



When the instrument is attached to the belt in an upright 
position, as here shown, the jar given it at each step causes the 
weighted lever shown in Fig. 31 to drop upon the adjustable 
screw 5. The lever recovers its position by the aid of a spring, 
and in so doing turns a ratchet-wheel by an amount propor- 
tional to the amplitude of the lever's motion. This may be 
adjusted to any length of pace by means of the screw S, which 
is turned by a key.,, The face is graduated like that of a watch, 
and gives the distance travelled in miles. This instrument 
may also be used on a horse, and when adjusted to the length 
of a horse's step will give equally good results. The accuracy 
of the result is in proportion to the uniformity of the steps, 
after having been adjusted properly for a given individual. 
The instrument is only used on explorations, preliminary sur- 
veys, and reconnoissance-work. 

The Length of Mens Steps has been investigated by Prof. 
Jordan,* of the Hanover Polytechnic School. From 256 
step-measurements by as many different individuals, of lines 
from 650 to 1000 feet in length, carefully measured by 
rods and steel tapes, he concludes that the average length oi 
step is 2.648 feet, ranging from 2.066 to 3.182 feet. The mean 
deviation from this amount for a single measurement was 
± 0.147 feet, or 5f- per cent. The average age of the persons 
making these step-measurements was 20 years. The length of 
step decreases with the age of the individual after the age of 
25 to 30 years. It is also proportional to the height of the 
person. The results for 18 different-sized persons gave the 
following averages : 



Height of person 


5'-o8 
2 .46 


5'.2 5 


5'4i 


5'. 5 8 


S-.74 


5'-9° 


6'.o7 


6'.23 

2 .76 


6'40 
2.79 


6'.56 


Length of step. • 


2.53 


2.56 


2-59 


2 .62 


2 .69 


2 .72 


2.85 



* See translation in Engineering News and American Contract Journal for 
July 25, 1885. 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 1 39 

On slopes the step is always shorter than on level ground, 
whether one goes up or down. The following averages from 
the step-measurement of 136 lines on mountain-slopes along 
trails were found : 





o° 


5° 


IO° 


15° 


20° 
I '.64 


25° 

i'48 


30° 

l'.25 






2'-53 


2'. 30 


2'.03 


i'.84 




2'-53 


2'43 


2'. 36 


2'.30 


2'. 20 


i'.97 


I '.64 



The length of the step is also found to increase with the 
length of the foot. One steps farther when fresh than when 
tired. The increase in the length of the step is also in nearly 
direct proportion to the increase of speed in walking. 

When the proper personal constants are determined, and 
when walking at a constant rate, distances can be determined 
by pedometer, or by counting the paces, to within about two 
per cent of the truth. One should always take his natural step, 
and not an artificial one which is supposed to have a known 
value, as three feet, for instance. Let a base be measured off 
and each student determine the length of his natural step when 
walking at his usual rate, or, what is the same thing, find how 
many paces he makes in 100 feet. He then has always a 
ready means of determining distances to an approximation, 
which in many kinds of work is abundantly sufficient. 



THE ODOMETER. 

140. The Odometer is an instrument to be attached to 
the wheel of a vehicle to record the number of revolutions 
made by it. One form of such an instrument is shown in 
Fig. 32 attached to the spokes of a wheel. 

Each revolution is recorded by means of the revolution of 
an axis with reference to the instrument, this axis really being 



140 



SURVEYING. 



held stationary by means of an attached pendulum which does 
not revolve. The instrument really revolves about this fixed 
axis at each revolution of the wheel, and the number of times 




Fig. 32. 

it does this is properly recorded and indicated by appropriate 
gearing and dials. 

This method of measuring distances is more accurate than 
by pacing, as the length of the circumference of the wheel is a 
constant. This length multiplied by the number of revolu- 
tions is the distance travelled. It is mostly used by exploring 
parties and in military movements in new countries which have 
not been surveyed and mapped. 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 141 



THE CLINOMETER. 

141. The Clinometer is a hand-instrument for determin- 
ing the slope of ground or the angle it makes with the horizon. 
It consists essentially of a level bubble, a graduated arc, and a 
line of sight, so joined that when the line of sight is at any angle 
to the horizon the bubble may be brought to a central position 
and the slope read off on the graduated arc. Such a combina- 
tion is shown in Fig. 33. It is called the Abney level and 




Fig. 33. 

clinometer, being really a hand-level when the vernier is set to 
read zero. The position of the bubble is visible when looking 
through the telescope, the same as with the Locke hand-level, 
shown in Fig. 16, p. 82. The body of the tube is made square, 
so that it may be used to find vertical angles of any surface by 
placing the tube upon it and bringing the bubble to the centre. 
The graduations on the inner edge of the limb give the slope 
in terms of the relative horizontal and vertical components of 
any portion of the line ; thus, a slope of 2 to I signifies that 
the horizontal component is twice the vertical. In reading this 
scale the edge of the vernier-arm is brought into coincidence 
with the graduation. 

This instrument is very useful in giving approximate slopes 
in preliminary surveys, the instrument being pointed to a posi- 



142 



SURVEYING. 



tion as high above the ground as its own elevation when held 
to the eye. 

THE OPTICAL SQUARE. 

142. The Optical Square is a small hand-instrument used 
to set off a right angle. It is shown in Fig. 34, the method of 




its use being evident from the figure. Thus, while the rod at 
is seen directly through the opening, the rod at p is seen in 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 143 



the glass as the prolongation downwards of that of o y it being 
reflected from the mirrors /"and c in succession, they having 
an angle of 45 with each other. By this means a line may be 
located at right angles to a given line at a given point, or a 
point in a given line may be found in the perpendicular to this 
line from a given point. 

THE PLANIMETER. 

143. The Planimeter is an ingenious instrument used for 
measuring irregular areas. It is a marked example of high 
mathematical analysis embodied in a very simple and useful 
mechanical appliance. Many forms of it are now in use, three 




Fig. 35. 

of the best of which will be described. The instrument has 
come to be a necessity in all kinds of surveying and engineer- 
ing work where irregular areas have to be evaluated. It is 
important that the student should thoroughly understand its 
principles, that he may use it with the greatest efficiency. 
The demonstration of its competency to measure areas is 
necessarily somewhat involved, and requires a little patient 
consideration. The demonstrations here given, though fol- 
lowing the methods of the calculus, are free from the peculiar 
notation there used. The form of the instrument shown in 
Fig. 35 is known as Amsler's Polar Planimeter. The point e 
is fixed by means of a needle-point puncturing the paper. The 
point d is made to pass over the perimeter of the area to be 
measured, and the record given by the rolling-wheel c and the 



144 SURVEYING. 



record-disk / is the area in the unit for which the length of the 
arm h was set. The rolling-wheel is mounted on an axis 
which is parallel to the arm h, and moves with a minimum 
amount of friction. It is evident that any motion of the 
wheel c in the direction of its axis would not cause it .to re- 
volve, while any motion at right angles to this axis is fully 
recorded by the wheel. The arm ei is of fixed length, while 
the length of the arm h is adjustable. 

144. Theory of the Polar Planimeter.* — In Fig. 36 let C 
represent the point where the instrument is fastened to the 
paper, and CP the arm, of fixed length m, whose only motion 




Fig. 36. 

is that of revolution on C as a centre, causing P to move in a 
circular arc. RT is the other arm, revolving on P as a centre, 
and carrying at the fixed distance RP (== ri) from Pa. rolling- 
wheel whose periphery touches the paper at R and whose axis 
is parallel to RP. RTalso carries at a distance TP (= /) from 
P the tracing-point, T\ I is a constant while the instrument is 
in use, though capable in the best instruments of having dif- 
ferent values given to it for different purposes. 

* The demonstration here given was published by Mr. Fred. Brooks, in the 
Journal of the Association of Engineering Societies, vol. iii. , p. 294, and is rep- 
resented as a joint production by himself and Mr. Frank S. Hart. A few slight 
changes and additions are here made. 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 145 

T and R can move nearer to or further from C only by the 
motion of the arm TR on P as a centre varying the angle X. 

The distance CT — |/(m + / cos Xf + (/ sin Xf 
= |/^ 2 + / 2 + 2;///cos X, 

as may be seen by dropping a perpendicular 7^ from T on cTP 
produced. 

To every particular value of X correspond particular values 
of CT, CR, angle CRP, etc. ; and successive small variations in 
X are accompanied by successive small variations in these 
quantities. When T, starting at any given distance from C, is 
moved through any path to the same or another place equally 
distant from C, the usefulness of the instrument depends upon 
X's coming back to its first value by passing in reverse order 
through the changes it has once made. This is secured by 
the usual construction of the instrument, which prevents T 
and R from crossing the line of CP; in other words, X, ex- 
pressed as arc to radius unity, is never less than nor more 
than 7t (a half-circumference). 

The only motion possible besides those above described is 
the turning of the rolling-wheel on its axis, which is produced 
by the component of the motion of R perpendicular to RP, 
that is, tangential to its periphery ; but the wheel does not 
turn for the component of the motion of R in the direction 
RP, which is parallel to the axis. Suppose, for simplicity, that 
the periphery of the wheel has a length of one unit and that 
the number of turns and fractions of a turn it makes upon any 
trial is recorded ; for, whatever the size and graduations may 
be, a simple calculation would reduce the results to the re- 
quired equivalents. To illustrate, let the arm RT turn on P 
as a centre, while CP remains fixed, from the position of the 
full line to that of the dotted line sk ; R moves to s, describing 
10 



14^ SURVEYING. 



an arc which is everywhere at right angles to its radius RP; 
hence the record of the wheel is the length of the arc Rs. On 

the other hand, supposing that CsP is a right angle ( — J and 

n 
cos CPs r= — , let both arms revolve around C with X fixed 
m 

equal to CPs ; the wheel is at every point moving parallel to 

its own axis, and its record is zero. The distance Ck of the 

tracing-point from C in this case may be found by substituting 

fi 
the value — for the cos X in the general expression for CT. 
m or 



which gives Vm* -\- T -\- 2nl. The circumference described by 
the tracing-point with this radius may be called the zero-cir- 
cumference. 

If both arms similarly revolved around (7 with X fixed at 
any other value between o and 7r, the axis of the wheel would 
make an oblique angle with the direction of R's path, and the 
wheel would partly roll and partly slip. The further CRP 

n 
varied from — , the less in proportion would be the slipping 

component, and the greater the rolling component and the 

record of the wheel. With CRP> — , T would describe an 

arc outside the zero-circumference and the wheel would make 

what we will call a positive record. With CRP < — , T would 

describe an arc inside the zero-circumference and the wheel 
would turn in the contrary direction, which we will call nega- 
tive ; provided that T revolved in the same direction in both 
cases. Motion of T through any arc in the direction of the 
hands of a watch may be considered positive ; then motion of 
T in the opposite direction over the same arc back to its start- 
ing-place must be considered negative, and would obviously be 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 1 47 

attended by a turning of the wheel equal in amount to that 
attending the positive movement, but with its direction re- 
versed. 

In the practical use of the instrument T may move over 
any path, near enough to the zero-circumference to be reached, 
whose beginning and end are equally distant from C. Hence 
X is the same at the end as at the beginning. The record 
thus made on the wheel is proportional to the area included 
between the zero-circumference and T's path and the radial 
lines through its beginning and end from the centre C, as will 
now be explained. 

T's path may be resolved into an infinite number of parts, 
consisting of infinitesimal arcs (/) described from Pas a centre 
by changes in X, CP being fixed, and of infinitesimal arcs (J) 
described from C as a centre with X fixed. This is illustrated 
by large arcs of the two classes on the diagram. The area in 
question may be correspondingly divided into elementary por- 
tions (illustrated by the large divisions made on the diagram 
by fine radial lines) each of which may be described as plus 
the area included between one of these infinitesimal arcs and 
radial lines through its extremities from C, minus the sector 
included by the same radii and an arc of the zero-circumfer- 
ence. Hence the area is a minus quantity if T moves inside 
the zero-circumference, positive if outside ; provided that T 
moves around C in the direction of the hands of a watch. If 
T moves around C in a contrary direction, both the signs in 
the above expression are to be changed ; for as the area of a 
sector is equal to its arc multiplied by half its radius, the area 
becomes negative when the arc becomes negative. If this be 
borne in mind it will be seen that the algebraic sum of all the 
elements corresponding to the second term in the above ex- 
pression is the sector of the zero-circumference included by 
radii passing through the points of beginning and ending of 



I48 SURVEYING. 



T's path ; and that the algebraic sum of all the elements 
corresponding to the first term is the area inclosed by 7"s 
path and lines from C to its beginning and end, however irreg- 
ular T's path may be. 

We will first consider that class of infinitesimal arcs (/) and 
corresponding elements of area, due to changes in X alone. 
Their accumulated effect upon both the area and upon the 
record of the rolling-wheel is zero. As to the wheel, from the 
condition that X passes again in reverse order through the 
changes it has once made, it follows that for every infinitesi- 
mal motion, like Rs, of R, recorded by the wheel for the 
infinitesimal change (/) between two consecutive values of X, 
there must be in some other place a motion in the opposite 
direction of the same magnitude for the infinitesimal change 
back again between two consecutive values of X equal to the 
former pair. As to the area, each infinitesimal arc / (like Tk) 
has, as previously stated, its corresponding element of area ; 
and the equally large arc with the contrary sign, just now 
referred to, in another place where X has the same values, 
must also have its corresponding element of area, exactly as 
large as the former, but with its algebraic sign reversed. The 
effect of the first class of elements into which ^'s path was 
resolved is thus eliminated. 

Hence the total record of the wheel for T's whole path is 
the record due to the second class of its elements, the infini- 
tesimal arcs {J) described from C with X fixed for each ; and 
the total area included between the zero-circumference, T's, 
path, and the terminal radii is the sum of all the elements of 
area corresponding to this second class of arcs (J), which we 
have now to consider. J expresses in terms of arc to radius 
unity any infinitesimal angle TCf between radial lines passing 
from C through the extremities of an infinitesimal arc Tf. 
The corresponding element of area is the difference between 
the sector TfC and the sector included by the zero-circumfer- 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 1 49 

ence and the same radii. Making use of the algebraic expres- 
sions given above, 

from %J(m 2 + / 2 + 2 {ml cos. X) 
subtract \ J(n? + I 2 + 2nl) 
and the difference J I (in cos. X — ri) 

is the required element of area. 

The corresponding record of the wheel is made by the 
motion of R through the path Re =J X CR. This path may 
be resolved into two components, Rh, which has no effect 
upon the record, and he, which is the record =/x CR X cos 
(n — CRP). By dropping the perpendicular Cg upon PR 
produced it will be seen that CR cos {n — CRP) = Rg = 
m cos X — n. Hence record of wheel is J X (m cos X — ri). 
Vherefore the element of area corresponding to an infinitesimal 
arc, J, is just I times the record due to the same arc ; hence the 
sum of the elements of area for all the arcs {J) is / times the 
total record corresponding, which is the essential thing that 
was to be proved. 

In the application of the instrument to get the area of a 
closed figure, T's path ends in the same point where it began, 
and we have two cases according as this is accomplished by 
CP's making a complete revolution around C, or by its mov- 
ing backward as much as it has once moved forward. In 
the first case, C is within the figure ; in the second, outside. 
In both cases the area between T's path and the terminal 
radii is the area of the closed figure. The sector within the 
zero-circumference, which we have been deducting, is in the 
first case the whole circle n (m* -f- P -\- 2nl) ; in the second, 
nothing. Hence add n (m 2 '-\- P -\- 2nl) to / times the record 
in the first case, and add nothing to it in the second, in order 
to get the required area of the closed figure. 



1 50 SUK VE YING. 



To show that the proper summation' is made on the wheel 
for the areas outside and inside the zero-circle, 

let A == area generated by the line ^77^when the point 

T is outside of circle; 
" A t = area generated by the line CT when the point 

T is inside of circle ; 
" 5= area of sector between radii to points where 

the perimeter crosses the zero-circle ; 
" A = area of the figure. 
Then A — S = outer area, and 
S — Ai = inner area. 

The sum of these is 

A = (A -S) + (S-A i ) = A -A i . 

But since A t is recorded negatively on the wheel, while A 
is recorded positively, the wheel record is 

A — ( — Ai) = A -\- Ai = A. 

145. To find Length of Arm to give area in any desired 
unit. In the previous article it was shown that the area was 
always / times the wheel record, where / was the length of the 
arm carrying the tracing-point, or the distance PT in Fig. 36. 
The wheel record is evidently its net circumferential move- 
ment, or Hc\ where 

n = number of revolutions of wheel shown by the differ- 
ence between the initial and final readings, 
and c = circumference of wheel. 
We may then write for the area of the figure 

A = Inc. 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 1 5 1 

If / and c are given in inches A will be found in square inches, 
and the same for any other unit. To cause an area of I square 
inch to register I revolution of the wheel, we will have 

I = Ic , 

or, / = — • 

c 

If c were 2 inches, this would give /= \ inch, which would 
be too short for practical purposes. Let us assume, then, that 
I square inch shall be registered as o.i revolution of the 
wheel. Then we have 

i = o.i Ic, 

7 IO 

or, / — — . 

c 

On an instrument the author has used c = 2.347 inches, 
whence for o.i revolution to correspond to 1 square inch area 
we have 

/= = 4.26 inches. 

2.347 

When this length of arm is carefully set off by the appro- 
priate clamp- and slow-motion screw, the area is given in 
square inches by multiplying the number of revolutions of the 
wheel by 10. A vernier is provided for reading the revolu- 
tions of the wheel to thousandths ; hence if it be read to 
thousandths, and two figures pointed off, the result is the area 
of the diagram moved over in square inches. 

It is evident that c can be evaluated in centimetres, and the 
corresponding metrical length of / found for giving the result 
in the metric notation. The exact circumference of the wheel 
is determined by the makers, and remains a constant for that 



152 



SURVEYING. 



individual instrument, giving a corresponding set of values 
of /. Since no two instruments are likely to have exactly the 
same wheel-circumference, so the settings for one instrument 
cannot be used for another. 

It must be kept in mind that the result is given in absolute 
units of area of the diagram, and this result must then be 
evaluated according to the significance of such unit on the 
diagram. Thus, if a sectional area has been plotted with a 
vertical scale of io feet to the inch and a horizontal scale of 
ioo feet to the inch, then one square inch on this diagram rep- 
resents iooo square feet of actual sectional area. The number 
of square inches in the figure as given by the planimeter must 
then be multiplied by iooo to give the area of the section in 
square feet. 

146. The Suspended Planimeter.— This is shown in Fig. 
37, It is essentially a polar planimeter, the pole being at C. 




Fig. 37. 



It has the advantage of allowing the wheel to move over the 
smooth surface of the plate S, instead of over the paper, thus 
giving an error about one sixth as great as that of the ordina- 
ry polar instrument. The theory of its action is essentially 
the same as the other. 

147. The Rolling Planimeter is the most accurate instru- 
ment of its kind yet devised. Its compass is also indefinitely 
increased, since it may be rolled bodily, over the sheet for any 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 1 53 

distance, on a right line, and an area determined within certain 
limits on either side. It is therefore especially adapted to the 
measuring of cross-sections, profiles, or any long and narrow 
surface. Fig. 38 shows one form of this instrument as de- 
signed by Herr Corradi of Zurich. It is a suspended planim- 
eter, inasmuch as the wheel rolls on a flat disk which is a 
part of the instrument, but it could not be called a polar pla- 
nimeter, the theory of its action being very different from that 
instrument. The frame B is supported by the shaft carrying 




Fig. 38. 



the two rollers R x . To this frame are fitted the disk A and the 
axis of the tracing-arm F. The whole apparatus may thus move 
to and fro indefinitely in a straight line on the two rollers while 
the tracing-point traverses the perimeter of the area to be 
measured. The shaft carries a bevel-gear wheel, R„ which 
moves the pinion R z . This pinion is fixed to the axis of the 
disk, and turns with it, so that any motion of the rollers R l 
causes the disk to revolve a proportional amount, and the 
component of this motion at right angles to the axis of the 
wheel E is recorded on that wheel. If the instrument remains 



*54 



SURVEYING. 



stationary on the paper (the rollers R not turning) and the 
tracing-point moved laterally, it will cause no motion of the 
wheel, since its axis is parallel to the arm F, and turns about 
the same axis with F, but 90 from it ; the wheel £, therefore, 
moves parallel with its axis and does not turn. 

148. Theory of the Rolling Planimeter. — This will be 
developed by a system of rectangular coordinates, the path of 
the fulcrum of the tracing-arm being taken as the axis of 





Fig. 39. 

abscissae. The path of the tracing-point will be considered 
as made up of two motions, one parallel to the axis of abscis- 
sae and the other at right angles to it. The elementary area 
considered will be that included between the axis of abscissae 
and two ordinates drawn to the extremities of an elementary 
portion of the path. But since this element of the perimeter 
is supposed to be made up of two right lines, one perpendicu- 
lar to the axis of abscissae and the other parallel to it, our 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 1 55 

elementary area must also be divided in a similar manner. 
It will at'once be seen that one part of this area is zero, since 
the two ordinates bounding it form one and the same line. 
This is the part generated by the motion at right angles to 
the axis of abscissae. Now, we have just shown in the previous 
article that the wheel-record for this part of the path is also 
zero. We are brought therefore to this important conclusion : 
that all components of motion of the tracing-point at right angles 
to the axis of abscisses have no influence upon the result. We 
will therefore only discuss a differential motion of the tracing- 
point in the direction of the axis of abscissae. 

In Fig. 39, which is a linear sketch of the instrument shown 
in Fig. 38, with the corresponding parts similarly lettered, it 
is to be shown that the motion of the wheel E caused by the 
movement of the tracing-point over the path dx is equal to 
the corresponding area ydx multiplied by some constant which 
is a function of the dimensions of the instrument. 

It is evident that a motion of the tracing-point in the di- 
rection of the axis of abscissae can only be obtained by moving 
the entire instrument on the rollers by the same amount, and 
therefore when the point moves over the path dx the circum- 
ferences of the rollers R 1 have moved the same amount. This 

causes a movement of the pitch circle of R 9 of dx -^. This 

motion is conveyed to the disk through R 3 , so that any point 
on this disk, as a, distant ad from the axis, moves through a 

distance equal to dx -^ - -_-. Let ab y Fig. 39, be this distance, 

Then we have 

R^ ad 
ab = dx-j=r - -rr (i) 



R, R 



The motion of that portion of the disk on which the roller 
rests, equal to a&, causes the circumference of the wheel E to 



156 SURVEYING. 



revolve by an amount equal to the component of the distance 
ab perpendicular to the axis of the wheel. This component 
part of the disk's motion is be, and this is the measure of the 
wheel's motion. It therefore remains to show that bc=ydx 
multiplied by an instrumental constant. 

Now, be = ab sin bac (2) 

But bac = a -f- /?, since gac and bad are both right angles. 
Also, bac = supplement of dag— a-\- ft. 
Also, from the triangle dag, we have 

sin dag : sin agd : : D : ad, 

. , ^ N D sin a 
or sin (« + /?) = ad (3) 



Since Fga is also a right angle, we have the angle formed 
Fg and the axis of a. 
We may now write : 



y 

by Fg and the axis of abscissae equal to a, whence sin a = ~ m 



be — ab sin (a + /3) = ab — ™ * = ab pT^~d' ' ^ 
Now, substituting the value of ab from (1), we have 

cl "= ■■*** F ■ M x - & . ••;••• (5) 

Since D, R 2 , F, R v and ^ 3 are all constants for any one 
instrument, we see that the wheel-record is a function of the 
area generated by the tracing-point and the instrumental con- 
stants, which was to be shown. It now follows that the sum- 
mation of all these elementary areas included between the 
path of the tracing-point, the limiting ordinates, and the axis 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 1 57 

of abscissae, is represented by the total wheel-movement , or the 
difference between its initial and final readings. If, therefore, 
the area to be measured is of this character, being bounded by- 
one right line and limiting ordinates, it would not be necessary 
to move the point over the entire perimeter, but only along 
the irregular boundary, provided the instrument could be ad- 
justed with the point g exactly over the base of the figure, and 
with the axis B at right angles to it, so that in rolling the in- 
strument along, the point g would remain over the base-line. 
In other words, the axis of abscissae of the instrument would 
have to coincide exactly with this base-line. Then for motion 
of the tracing-point over this line, as well as for its motion over 
the end-ordinates, the wheel would not revolve, neither would 
there be any area generated between these lines and the axis. 
In general this cannot be done, and it is only mentioned here 
in order to more clearly illustrate the working of the instru- 
ment. 

As in the case of the polar instrument, the proper length of 
arm F, to be used with the rolling-planimeter to give results 
in any desired unit, depends on the other instrumental con- 
stants. These being known, the value of F may be computed 
in the same manner as with the polar planimeter. 

149. To test the Accuracy of the Planimeter, there is 
usually provided a brass scale perforated with small holes. A 
needle-point is inserted in one of these and made fast to the 
paper or board, while the tracing-point rests in another. The 
latter may now be moved over a fixed path with accuracy. 
Make a certain number of even revolutions forward, or in the 
direction of the hands of a watch, noting the initial and final 
readings. Reverse the motion the same number of revolutions, 
and see if it comes back to the first reading. If not, the dis- 
crepancy is the combined instrumental error from two meas- 
urements due to slip, lost motion, unevenness of paper, etc. 

If this test be repeated with the areas on opposite sides of 



158 SURVEYING. 



the zero-circle in the case of the polar-planimeter, or on oppo- 
site sides of the axis of abscissae in case of the rolling-planime- 
ter, with the same score in both cases, it proves that the pivot- 
points a, b, k, and the tracing-point d (Fig. 35), are in the same 
straight line, in case of the polar instrument, and that the cor- 
responding points in the suspended and rolling planimeters 
form parallel lines ; in other words, that the axis of the meas- 
uring-wheel is parallel to the tracing-arm. If the results differ 
when the areas lie on opposite sides of the axis or zero-circle, 
these lines are not parallel and must be adjusted to a parallel 
position. 

150. Use of the Planimeter. — The paper upon which the 
diagram is drawn should be stretched smooth on a level sur- 
face. It should be large enough to allow the rolling-wheel to 
remain on the sheet. 

The instrument should be so adjusted and oiled that the 
parts move with the utmost freedom but without any lost mo- 
tion.. This requires that all the pivot-joints shall be adjustable 
to take up the wear. The rim of the measuring-wheel must be 
kept bright and free from rust. The instrument must be han- 
dled with the greatest care. Having set the length of the 
tracing-arm for the given scale and unit, it is well to test it 
upon an area of known dimensions before using. If it be found 

to give a result in error by — of the total area, the length of the 

tracing-arm must be changed by an amount equal to this same 
ratio of its former length. If the record made on the wheel 
was too small then the length of the tracing-arm must be di- 
minished, and vice versa. If the paper has shrunk or stretched, 
find the proportional change, and change the length of the 
tracing-arm from its true length as just found, by this same 
ratio, making the arm longer for stretch and shorter for shrink- 
age. Or the true length of arm may be used, and the results 
corrected for change in paper. 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 1 59 

To measure an area, first determine whether the fixed point, 
or pole, shall be inside or outside the figure. It is preferable 
to have it outside when practicable, since then the area is ob- 
tained without correction. If, however, the diagram is too 
large for this (in case of the polar planimeter) the pole may be 
set inside. In either case inspection, and perhaps trial, is nec- 
essary to fix upon the most favorable position of the pole, so 
that the tracing-point may most readily reach all parts of the 
perimeter. If the area is too large for a single measurement, 
divide it by right lines and measure the parts separately. 
Having fixed the pole, set the tracing-point on a well-defined 
portion of the perimeter, and read and record the score on 
the rolling-wheel and disk. This is generally read to four 
places. Move the tracing-point carefully and slowly over the 
outline of the figure, in the direction of the hands of a watch, 
around to the initial point. Read the score again. If the 
pole is outside the figure, this result is always positive when 
the motion has been in the direction here indicated. If the 
pole is inside the figure, the result will be negative when the 
area is less than that of the zero-circle, positive if greater. 
With the pole inside the figure, however, the area of the zero- 
circle must always be added to the result as given by the score, 
and when this is done the sum is always positive, the motion 
being in the direction indicated. The area of this zero-circle 
is found in art. 144, to be n (m* -f- P -f- 2nl). The value 
of /, which is the length of the tracing-arm, is known. The 
values of m and n should be furnished by the maker. If these 
are unknown, the area of the zero-circle can be found for any 
length of arm /, by measuring a given area with pole outside 
and inside, the difference in the two scores being the area of 
this circle. By doing this with two very different values of / 
we may obtain two equations with two unknown quantities, 
m and n, from which the absolute values of these quantities 
may be found. Thus we would have: 



l6o SURVEYING. 



A = n (m* +/ 2 + 2nl) ; 
A' , = ?r(« 1 + r+2«/ / ); 



^4 
whence W + 2 ?^ = ^ 2 ; 

7t 



7t 



wherein /, /', A> and A' are known. The values of m and # are 
then readily found. 

In using the rolling-planimeter, it is advisable to take the 
initial point in the perimeter on the axis of abscissae, as in this 
position any small motion of the tracing-point has no effect 
on the wheel, and so there is no error due to the initial and 
final positions not being exactly identical. 

The planimeter may be used to great advantage in the 
solution of many problems not pertaining to surveying. In 
all cases where the result dan be represented as a function 
of the product of two variables and one or more constants, the 
corresponding values of the variables may be plotted on cross- 
section paper by rectangular coordinates, thus forming with 
the axis and end-ordinates an area which can be evaluated for 
any scale and for any value of the constant-functions by setting 
off the proper length of tracing-arm. Thus, from a steam- 
indicator card the horse-power of the engine may be read off, 
and from a properly constructed profile the amount of earth- 
work in cubic yards in a railway cut or fill. Some of these 
special applications are further explained in Part II. of this 
work. 

151. Accuracy of Planimeter-measurements. — Professor 
Lorber, of Loeben, Austria, has thoroughly investigated the 
relative accuracy of different kinds of planimeters, and the re- 
sults of his investigations are given in the following table. It 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. l6l 



will be seen that the relative error is less as the area measured 
is larger. The absolute error is nearly constant for all areas, in 
the polar planimeter. The remarkable accuracy of the rolling- 
planimeter is such as to cause it to be ranked as an instrument 
of precision. 

TABLE OF RELATIVE ERRORS IN PLANIMETER-MEASUREMENTS. 



Area 


IN — 


The error in one passage of the tracer amounts on an 
average to the following fraction of the area meas- 
ured by — 




The ordinary po- 
lar planimeter- 
Unit of vernier: 
10 sq. mm. 
= .015 sq. in. 


Suspended plani- 
meter -Unit of 
vernier: 
1 sq. mm. = 
.001 sq. in. 


Rolling planime- 


Square cm. 


Square inches. 


ter-Unit of ver- 
nier: 
1 sq. mm. = 
.001 sq. in. 


IO 


1-55 


1 


1 


TDTHF 


20 


3.IO 


Tk 


1 
TTTT 


-jinnr 


50 


' 7.75 


1 
T5S 


2500" 


TUTTG 


IOO 


I5-50 


1 


TIST 


BOOff 


200 


31-00 


- 1 ■ 
T2TT 


1 

TTT3 


T^5T 


300 


46.50 





1 


l 


S3»o 


TOOL'S 



THE PANTOGRAPH. 

152. The Pantograph is a kind of parallel link-motion 
apparatus whereby, with one point fixed, two other points are 
made to move in a plane on parallel lines in any direction. 
The device is used for copying drawings, or other diagrams to 
the same, a larger, or a smaller scale. The theory of the instru- 
ment rests on the following: 

Proposition : If the sides of a parallelogram, jointed at 
the corners A, B, C, and D, and indefin itely extended, be cnt by 
a right line in four points, as E, F, G, and H, then these latter 
points will lie in a straight line for all values of each of the 
parallelogram angles from zero to 180 , and the ratio of the dis- 
tances EF, FG, atid GH will remain unchanged. 



11 



1 62 



SURVEYING. 



In Fig. 40, let A, B, C, Dbe the parallelogram, whose sides 
(extended) are cut by a right line in F, G, E, and H. It is 
evident that one point in the figure may remain fixed while 




— yrTf^n-E o 



Fig. 40. 



the angles of the parallelogram change. Let this point be G. 
Since GC and GH, radiating from G, cut the parallel lines 
DE and CH, we have 

GD\DE :: GC : C/7. 

Also, for similar reasons, 

£Z>: Z>£ :: EA \ AF. 

Now since the sides of the parallelogram, as well as all the 
intercepts, AF, GD, DE, and CH, remain constant as the angles 
of the figure change, when the figure has taken the position 
shown by the dotted lines, we still have 



also, 



GD' : HE :: GC : C'H'\ 



EH' : HG :: E'A f : A'F. 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 1 63 



From the first of these proportions we know that G, E\ 
and H' are in the same straight line, and the same for G, E, 
and F ; therefore, they are all four in the same straight line. 

To show that they are the same relative distance apart as 
before we have, 

FG\GE\ EH :: BC : DE : CH-DE; 
also, 

EG : GE : E'H' :: B'C : D'E ! \ CH'-D'E'. 
But 

BC = B'C, DE = D'E, and CH - DE = £'#' - Z>'£'; 
therefore we may write, 

EG: GE: EH :: EG : ££' : £77'. 

Q.E.D. 

It is evident that two of the points E, E, G, and H may 
become one by the transversal passing through the point of 
intersection of two of the sides of the parallelogram. The 
above proposition would then hold for the three remaining 
points. 

In the Pantograph only three of the four points E, F, 
G, and H (Fig. 40) are used. One of these may therefore 
be taken at the intersection of two sides of the .parallelogram, 
but it is not necessarily so taken. These three points are : the 
fixed point, the tracing-point, and the copying-point. 

In Fig. 41, ./ms the fixed point, held by the weight P; B is 
the tracing-point, and D is the copying-point, or vice versa as 
to B and D. The parallelogram is E s G, B, H. The points 



164 



SURVEYING. 



F, B f and D must lie in a straight line, B being at the inter- 
section of two of the sides of the parallelogram. The points 
A, E, and C are supported on rollers. In Fig. 42, the fixed- 




Fig. 41. 



point is the point of intersection of two of the sides of the 
parallelogram. The upper left-hand member of the frame is 
not essential to its construction, serving simply to stiffen the 




Fig. 4a. 



copying-arm, the fourth side of the parallelogram being the 
side holding the tracing-point. 

In Fig. 43, neither of the three points is at the intersection 
of two of the sides of the parallelogram, and hence there is a 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 1 65 

fourth point unused, having the same properties as the fixed, 
tracing, and copying points, it being at the intersection of the 
line joining these three points with the fourth side of the par- 
allelogram. 

From the theoretical discussion, and from the figures shown, 
it becomes evident that there may be an indefinite variety of 




Fig. 43. 

combinations which will do the work of a pantograph. The 
only essential conditions are that the fixed, the tracing, and the 
copying points shall lie in a straight line on at least three sides 
of a jointed parallelogram, either point serving any one of the 
three purposes. 

i-5?>* Use of the Pantograph. — The use of the instrument 
is easily acquired. Since both the tracing and copying points 
should touch the paper at all times, such a combination as that 
shown in Fig. 41 is preferable to those shown in Figs. 42 and 
43, since in these latter the tracing point is surrounded by sup- 
ported points, and so would not touch the paper at all times 
unless the paper rested on a true plane. In most instruments 
where the scale is adjustable, the two corresponding changes 
in position of tracing and copying points for different scales 
are indicated. To test these marks, see that the adjustable 
points are in a straight line with the fixed point, and to test the 

scale see that the ratio -™- (Fig. 41) is that of the reduction 



1 66 SURVEYING. 



desired. Thus, if the diagram is to be enlarged to twice the 
original size, make BD = 2FB; 

DE FE 
or make ^7=. = ^^ = scale of enlargement. 

DG BG & 

If the drawing is to be reduced in size, make B the copying- 
point and D the tracing-point. 

If the drawing is to be copied to the same scale, make BF 
= BD and make B the fixed point. The figure is then copied 
to same scale, but in an inverted position. 

In the best instruments the arms are made of brass, but 
very good work may be done with wooden arms. 

PROTRACTORS. 

154. A Protractor is a graduated circle or arc, with its cen- 
tre fixed, to be used in plotting angles. They are of various 
designs and materials. 

Semicircular Protractors, such as shown in Fig. 44, are 

usually made of horn, brass, or 
german-silver. They are grad- 
uated to degrees or half-degrees, 
and the angle is laid off by 
holding the centre at the vertex 
of the angle, with the plain 
edge, or the o and 180 degree 
IG * 44 ' line on the given line from which 

the angle is to be laid off. 

In the full circle protractor, shown in Fig. 45, there is a 
movable arm with a vernier reading to from I to 3 minutes. 
The horn centre is set over the given point, the protractor 
oriented with the zero of the circle on the given line, and the 
arm set to the given reading when the other line may be 
drawn. 




ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 1 67 

The three-arm protractor, Fig. 46, has one fixed and two 
movable arms by which two angles may be set off simulta- 
neously. It is used in plotting observations by sextant of two 




Fig. 45. 



angles to three known points for the location of the point of 
observation. This is known as the three-point problem and 
is discussed in Chap. X. 




Fig. 46. 



Paper protractors are usually full circled, from 8 to 14 
inches in diameter, graduated to half or quarter degrees. 
They are printed from engraved plates on drawing- or tracing- 



1 68 



SURVEYING. 



paper or bristol-board, and are very convenient for plotting 
topographical surveys. The map is drawn directly on the 
protractor sheet, the bearing of any line being taken at once 
from the graduated circle printed on the paper. These " pro- 
tractor sheets" can now be obtained of all large dealers. 

The coordinate protractor * is a quadrant, or square, with 




Fig. 47. 

angular graduations on its circumference, or sides, and divided 
over its face by horizontal and vertical lines, like cross-section 
paper. A movable arm can be set by means of a vernier to 
read minutes of arc, this arm being also graduated to read 
distances from the centre outward. Having set this arm to 
read the proper angle, the latitude is at once read off on the 



* Called a Trigonometer by Keuffel & Esser, the makers. 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 1 69 

vertical scale and the departure on the horizontal scale for the 
given distance as taken on the graduated arm. A quadrant 
protractor giving latitudes and departures for all distances 
under 2500 feet to the nearest foot, or under 250 feet to the 
nearest tenth of a foot, has been used. The radius of the cir- 
cle is i8f inches. Both the protractor and the arm are on 
heavy bristol-board, so that any change due to moisture will 
affect both alike and so eliminate errors due to this cause. 
The instrument was designed to facilitate the plotting of the 
U. S. survey of the Missouri River.* It has proved very 
efficient and satisfactory. A similar one on metal, shown in 
Fig. 47, is now manufactured, and serves the same purpose. 

PARALLEL RULERS. 

155. The Parallel Ruler of greatest efficiency in plotting 
is that on rollers, as shown in Fig. 48. The rollers are made 
of exactly the same circumfer- | 
ence, both being rigidly attached ^jp^ t r ___ 5JEj 

to the same axis. It should be ' * , , ,„„, 

made of metal so as to add to its FlG - * 8 - 

weight and prevent slipping. It is of especial value in connec- 
tion with the paper protractors, for the parallel ruler is set on 
any given bearing and then this transferred to any part of the 
sheet by simply running the ruler to place. Two triangles 
may be made to serve the same purpose, but they are not so 
rapid or convenient, and are more liable to slip. The parallel 
ruler is also very valuable in the solution of problems in 
graphical statics. 

SCALES. 

156. Scales are used for obtaining the distance on the 
drawing or plot which corresponds to given distances on the 

* For sale by A. S. Aloe & Co., St. Louis, Mo. 



170 



SURVEYING. 



object or in the field. There is such a variety of units for 
both field and office work, and a corresponding variety of 
scales, that the choice of the particular kind of scale for any 
given kind of work needs to be carefully made. Architects 
usually make the scale of their drawings so many feet to the 
inch, giving rise to a duodecimal scale, or some multiple of ^ 





6 , „ , 1 ,, , 


15 


4 




3 l 


2 i 


1 1 


J 1,1 1,1. .1.1 1 1 


MM 


MM 


miImii 


M M M M 1 


M II M 1 1 


llo 


9 


9 


7 


6 


5 


4- 


3 


i 


i 


2 4 Is 8 | 






































'tminil 






































' 1 M Ic 






































1 III 






































i Ilk 


'["''I'll 




































I u i ii ip 


4 H 1 1 > i > > 1 1 9 




































II, 






































ni e 






































1 ' 1 1 1 I n 


2 In 1 




































1 < I 1 1 1 1° 






































1 llll 1 II II 


1 1 1 1 


2 


3 


4 




6 


7 


sl 9 


1 





1 


z 


1 


4r 


\ 


e 


i 


a 


1 





Fig. 49. 



A surveyor who uses a Gunter's chain 66 feet in length plots 
his work to so many chains to the inch, making a scale of 
some multiple of 7 9 ^ . An engineer usually uses a 100-foot 
chain and a level rod divided to decimal parts of a foot ; so he 
finds it convenient to use a decimal scale for his maps and 
drawings, reduced to the inch-unit however. Here the field- 
unit is feet and the office-unit is inches, both divided deci- 
mally. This gives rise to a sort of decimal-duodecimal system, 
the scale being some multiple of 'y^r- Various combinations 
of all these systems are found. 

Figure 49 shows one form of an ivory scale of equal parts 
for the general draughtsman. The lower half of the scale is 
designed to give distances on the drawing for 4, 40, or 400 
units to the inch when the left oblique lines and bottom 
figures are used, and for 8, 80, or 800 units to the inch when 
the right oblique lines and top figures are used. Thus, if we 
are plotting to a scale of 400 feet to the inch, and the dis- 
tance is 564 feet, set one point of the dividers on the vertical 
line marked 5, and on the fourth horizontal line from the bot- 
tom. Set the other leg at the intersection of the sixth inclined 



ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. I/I 

line with this same horizontal line, and the space subtended by 
the points of the dividers is 564 feet to a scale of ^Vo- 

Figure 50 is a cut of an engineer's triangular boxwood 
scale, 12 inches long, being divided into decimal inches. 
There are six scales on this rule, a tenth of an inch being sub- 
divided into I, 2, 3, 4, 5, and 6 parts, making the smallest 



wiHjww\\\\\m\\\\\\m\™^^ 



Fig. 50. 



graduations £>, ^ £>, ^, 3W0 of an inch respectively. ^ This 
is called an engineer's or decimal-inch scale The architect's 
triangular scale is divided to give |, J, f, |, j, I, ih 2, 3, and 
4 inches to the foot. Such a scale is of little service to the 
civil engineer. 



BOOK II. 

SURVEYING METHODS. 



CHAPTER VII. 
LAND-SURVEYING. 

157. Land-surveying includes laying-out, subdividing, 
and finding the area of, given tracts of land. In all cases the 
boundary- and dividing-lines are the traces of vertical planes 
on the surface of the ground, and the area is the area of the 
horizontal plane included between the bounding vertical 
planes. In other words, the area sought is the area of the 
horizontal projection of the real surface. 

158. In laying out Land the work consists in running 
the bounding- and dividing-lines over all the irregularities of 
the surface, leaving such temporary and permanent marks as 
the work may demand. These lines to lie in vertical planes, 
and their bearings and horizontal distances to be found. The 
bearing of a line is the horizontal angle it makes with a merid- 
ian plane through one extremity, and its length is the length 
of its horizontal projection. This reduces the plot of the work 
to what it would be if the ground were perfectly level. If all 
the straight lines of a land-survey lie in vertical planes, and if 
their bearings and horizontal lengths are accurately deter_ 
mined, then as a land survey it is theoretically perfect, what- 
ever the purpose of the survey may be. 



SURVEYING METHODS. 1 73 



THE UNITED STATES SYSTEM OF LAYING OUT THE PUBLIC 

LANDS. 

159. The Public Lands of the United States have 
included all of that portion of our territory north of the Ohio 
River and west of the Mississippi River not owned by indi- 
viduals previous to the dates of cession to the United States 
Government. All of this territory, except the private claims, 
has been subdivided, or laid out, in rectangular tracts bounded 
by north and south and east and west lines, each tract having 
a particular designation, such that it is impossible for the pat- 
ents or titles, as obtained from the Government, to conflict. 
This has saved millions of dollars to the land-owners in these 
regions by preventing the litigations that are common in the 
old colonial States, and is one of the greatest boons of our 
national Government. The system was probably devised by 
Gen. Rufus Putnam,* an American officer in the Revolution- 
ary War. It was first used in laying out the eastern portion 
of the State of Ohio, in 1786-7, then called the Northwest 
Territory. This was the first land owrted and sold by the 
national Government. The details of the system have been 
modified from time to time, but it remains substantially un- 
changed. The following is a synopsis of the method which is 
given in detail in the Instructions to Surveyors-General, issued 
by the Commissioner of the General Land Office, at Washing- 
ton, D. C, and obtained on application. 

160. The Reference-lines consist of Principal Meridians 
and Standard Parallels. The principal meridians may be a 
hundred miles or more apart, but the standard parallels are 24 
miles apart north of 35 north latitude, and 30 miles apart 
south of that line. These lines should be run with great care, 

* See article by Col. H. C. Moore in Journal of the Association of En- 
gineering Societies, vol. ii., p. 282. 



174 SURVEYING. 



using the solar compass or solar attachment. The magnetic 
needle cannot be relied on for this work, for two reasons : there 
may be local attraction from magnetic deposits, and the dec- 
lination changes rapidly (about a minute to the mile) on east 
and west lines. The transit alone might be used to run out 
the meridians, as this consists simply of extending a line in a 
given direction. The transit could not be used for running the 
parallels, however, for these are ever changing their direction, 
since they are at all points perpendicular to the meridian at 
that point. This change in direction is due to the convergence 
of the meridians. The solar compass is the only surveying in- 
strument that can be used for running a true east and west 
line an indefinite distance. The needle-compass would do if 
there were no local attraction and if the true declination were 
known and allowed for at all points. The solar compass (or 
solar attachment) is the instrument recommended for this 
work. 

In running these reference-lines, every 80 chains (every mile) 
is marked by a stone, tre*e, mound, or other device, and is 
called a " section corner." Every sixth mile has a different 
mark, and is called a " township corner." 

161. The Division into Townships. — From each " town- 
ship corner" on any standard parallel auxiliary meridians are 
run north to the next standard parallel. Since these meridians 
converge somewhat towards the principal meridian, they will 
not be quite a mile apart when they reach the next standard 
parallel. But the full six-mile distances have been marked off 
on this parallel from the principal meridian, and it is from these 
township corners that the next auxiliary meridians will start 
and run north to the next standard parallel, etc. Thus each 
standard parallel becomes a " correction-line" for the merid- 
ians. The territory has now been divided into "ranges" 
which are six miles wide and twenty-four miles long, each range 
being numbered east and west from the principal meridian. 



SURVEYING METHODS. 1 75 

These ranges are then cut by east and west lines joining the 
corresponding township corners on the meridians, thus dividing 
the territory into " townships," each six miles square, neglect- 
ing the narrowing effect of the convergence of the meridians. 
The townships are numbered north and south of a chosen 
parallel, which thus becomes the " Principal Base-line." The 
fifth township north of this base-line, lying in the third range 
west of the principal meridian, would be designated as " town 
five north, range three west." Each township contains thirty- 
six square miles, or 23,040 acres. 

162. The Division into Sections. — The township is di- 
vided into thirty-six sections, each one mile square and contain- 
ing 640 acres. This is done by beginning on the south side of 
each township and running meridian lines north from the " sec- 
tion corners" already set, marking every mile or " section 
corner," and also every half-mile or " quarter-section corner." 
When the fifth section corner is reached, a straight line is run 
to the corresponding section corner on the next township line. 
This will cause this bearing to be west of north on the west, and 
east of north on the east, of the principal meridian. When this 
northern township boundary is a standard or correction-line, 
then the sectional meridians are run straight out to it, and thus 
this line becomes a correction-line for the section-lines as well 
as for the township-lines. The east and west division-lines are 
then run, connecting the corresponding section corners on the 
meridian section lines, always marking the middle, or quarter- 
section points. Evidently, to. run a straight line between two 
points not visible from each other, it is necessary first to run a 
random or trial line, and to note the discrepancy at the second 
point. From this the true bearing can be computed and the 
course rerun, or the points on the first course can be set over 
the proper distance. The sections are numbered as shown in 
Figs. 51 and 52. 

When account is taken of the convergence of meridians, the 



17$ SURVEYING. 



sections in the northern tiers of each township will not be quite 
one mile wide, east and west ; but as the section corners are set 
at the full mile distances on the township-lines, the southern 
sections in the next town north begin again a full mile in width. 
In setting the section and quarter-section corners on the east 
and west town lines the full distances are given from the east 
towards the west across each township, leaving the deficiency 
on the last quarter-section, or 40-chain distance, until the next 
correction-line is reached, when the town meridians are again 
adjusted to the full six-mile distances. 

163. The Convergence of the Meridians is, in angular 
amount,* 

c = m sin J (L -f- L f ) ; 

where ;// — meridian distance in degrees, or difference of longi- 
tude, and L and L' are the latitudes of the two positions. In 
other words, the angular convergence of the meridians is the 
difference in longitude into the sine of the mean latitude. 

The convergence in chains of two township-lines six miles 
apart, from one correction-line to another twenty-four miles 
apart, in lat. 40 , is 

C = 24 X 80 X sin c ; 

where c, in degrees, = -g^- sin 40 , since one degree of longitude 
in lat. 40 = 53 miles. Thus c = 4! .37 for each six-mile dis- 
tance, east or west, in lat. 40 . Whence C = 2.42 chains, 
which is what the northern tier of sections in the north range 
between correction-lines lacks of being six miles east and west. 
In a similar manner, we may find that the north sections in 
a town are about six feet narrower, east and west, than the 
corresponding southern sections in the same town. 

* See Chapter XIV. 



SURVEYING METHODS. 



1 77 



Figures 51 and 52 show the resulting dimensions of sections 
in chains when no errors are made in the field-work. The 
north and south distances are all full miles. 



Fig. 51. 



79.40 


80 


80 


80 


80 


80 


6 


5 


4 


3 


2 


1 


79.92 


79.92 


79.92 


79.92 


79.92 


79.92 


7 


8 


9 


10 


11 


12 


79-94 










79-94 


18 


17 


16 


15 


14 


13 


79-95 










79-95 


19 


20 


21 


22 


23 


24 


79-97 










79-97 


30 


29 


28 


27 


26 


25 


79.98 










79.98 


31 


32 


33 


34 


35 


36 


80 


80 


80 


80 


80 


80 



CORRECTION-LINE. 



- In Fig. 51 it will be observed that in the northern tier of 
sections the meridians must bear westerly somewhat so as to 
meet the full-mile distance, laid off on the township-line. 

In Fig. 52 they continue straight north to the town-line, 
which is in this case a correction-line. If the distances on this 
correction-line be summed they will be found to be 2.42 chains 
short of six miles as above computed. 

The law provides that all excesses or deficiencies, either 
12 



i;8 



SURVEYING. 



CORRECTION-LINE. 



78.08 


79.90 


79.90 


79.90 


79.90 


79.90 


6 


5 


4 


3 


2 


1 


78.10 










79.92 


7 


8 


9 


10 


11 


12 


78.12 










79-94 


18 


17 


16 


15 


14 


13 


78.13 










79-95 


19 


20 


21 


22 


23 


24 


78.14 










79-97 


30 


29 


28 


27 


26 


25 


78.16 










79.98 


3i 


32 


33 


34 


35 


36 


78.18 


80 


80 


80 


80 


80 



Fig. 52. 



from erroneous measurements or bearings or from the conver- 
gence of meridians, shall, so far as possible, be thrown into the 
northern and western quarter-sections of the township. 

164. Corner Boundaries have been established on all 
United States land surveys at the corners of townships, sec- 
tions, and quarter-sections, except at the quarter-section corner 
at the centre of each section. These corners have consisted 
of stones, trees, posts, and mounds of earth. Witness- or bear- 
ing-trees have always been blazed and lettered for the given 
town, range, and section, one tree in each section or town 
meeting at that corner, whenever such trees were available. 
The bearings and distances to such trees, and a description of 



SURVEYING METHODS. 1 79 

the same, are given in the field-notes. All such corners and 
witness points, except those made of stone, are subject to de- 
struction and decay, and when these are lost there is no means 
of relocating the boundary-lines. They were designed to serve 
only until the land should be sold off to individuals, when it 
was expected the owner would replace them with marks of a 
more permanent character. This has seldom been done, so 
that in many instances the sectional boundaries can now only 
be redetermined by personal testimony, line fences and other 
circumstantial evidence.* 

FINDING THE AREA OR SUPERFICIAL CONTENTS OF LAND 
WHEN THE LIMITING BOUNDARIES ARE GIVEN. 

165. The Area of a Piece of Land is the area of the level 
surface included within the vertical planes through the bound- 
ary-lines. This area is found in acres, roods, and perches, or, 
better, in acres only, the fractional part being expressed 
decimally. Evidently the finding of such an area involves two 
distinct operations, viz. : the Field-work, to determine the 
positions, directions, and lengths of the boundary-lines ; and 
the Computation, to find the area from the field-notes. There 
are several methods of making the field observations, giving 
rise to corresponding methods of computation. Thus, the 
area maybe divided into triangles, and the lengths of the sides, 
or the angles and one side, or the bases and altitudes measured, 
and the several partial areas computed. Or the bearings and 
distances of the outside boundary-lines maybe determined and 
the included area computed directly. This is the common 
method employed. Again, the rectangular coordinates of each 
of the corners of the tract may be found in any manner with 
reference to a chosen point which may or may not be a point 
in the boundary, and the area computed from these coordi- 
nates. These three methods will be described in detail. 

* See Appendix A. 



i8o 



SURVEYING. 



I. Area by Triangular Subdivision, 

166. By the Use of the Chain Alone. — In Fig. 
ABCDEF be the corner bound- 
aries of a tract of land, the sides 
being straight lines. Measure 
all the sides and also the diag- 
onals AC, AD, AE, and FB. 
The area required is then the 
sum of the areas of the four tri- 
angles ABC, ACD, ADE, and 
AEF. These partial areas are 
computed by the formula 



53 let 



Area = Vs(s — a)(s — b)(s — c), 




Fig. S3. 



where s is the half sum of the three sides a, b, c in each case. 

For a Check, plot the work from the field-notes. Thus, take 
any point as A and draw arcs of circles, with A as the com- 
mon centre, with the radii AB, AC, AD, AE, and AF taken to 
the scale of the plot. From any point on the first arc, as B, 
and with a radius equal to BC to scale, cut the next arc, whose 
radius was AC, giving the point C From C find D with the 
measured distance CD, etc., until F is reached. Measure FB 
on the plot, and if this is equal to the measured length of this 
line, taken to the scale of the drawing, the field-work and plot 
are correct. It is evident the point A might have been taken 
anywhere inside the boundary-lines without changing the 
method. 

167. By the Use of the Compass, or Transit, and Chain. 
— If the compass had been set up at A the outer boundaries 
could have been dispensed with, except the lines AB and AF. 
All that would be necessary in this case would be the bear- 
ings and distances to the several corners. We then have two 



SURVEYING METHODS. l8l 

sides and the included angle of each triangle given when the 
area of each triangle is found by the formula: 

Area = \ab sin C. 

In this case there is no check on the chaining or bearings. 
The taking-out of the angles from the given bearings could be 
checked by summing them. This sum should be 360 when 
A is inside the boundary-line, and 360 minus the exterior 
angle FAB when A is on the boundary. If the boundary- 
lines be measured also, then the area of each triangle can be 
computed by both the above methods and a check obtained. 

168. By the Use of the Transit and Stadia.*— Set up 
at A y or at any interior or boundary point from which all the 
corners can be seen, and read the distances to these corners 
and the horizontal angles subtended by them. The area is 
then computed by the formula given in the previous article. 
The distances may be checked by several independent read- 
ings, and the angles by closing the horizon (sum = 360 ). 

The above methods do not establish boundary-lines, which 
is usually an essential requirement of every survey. 



II. Area from Bearing and Length of the Boundary-lines. 

169. The Common Method of finding land areas is by 
means of a compass and chain. The bearings and lengths of 
the boundary-lines are found by following around the tract to 
the point of beginning. If the boundary-lines are unobstructed 
by fences, hedges, or the like, then the compass is set at the 
corners, and the chaining done on line. If these lines are ob- 
structed, then equal rectangular offsets are measured and the 

* The stadia methods are described in Chapter VIII. 



1 82 SURVEYING. 



bearings and lengths of parallel lines are determined. In this 
case the compass positions at any corner for the two courses 
meeting at that corner are not coincident, neither are the final 
point of one course and the initial point of the next course, 
the perpendicular offsets from the true corner overlapping on 
angles less than 180 and separating on angles over 180 . 

The chaining is to be done as described in art. 4, p. 8, the 
66-foot or Gunter's chain being used. Both the direct and the 
reverse bearing of each course should be obtained for a check 
as well as to determine the existence of any local attraction. 
For the methods of handling and using the compass see 
Chapter II. 

170. The Field-notes should be put on the left-hand page 
and a sketch of the line and objects crossing it on the right- 
hand page of the note-book. The following is a convenient 
form for keeping the notes. They are the field-notes of the 
survey which is plotted on p. 184. It will be seen that the 
" tree" was sighted from each corner of the survey and its 
bearing recorded. If these lines were plotted on the map 
they would be found to intersect at one point. If the plot 
had not closed, then these bearings would have been plotted 
and they would not have intersected at one point, the first 
line which deviated from the common point indicating that 
the preceding course had been erroneously measured, either in 
bearing or distance, or else plotted wrongly. In general such 
bearings, taken to a common point, enable us to locate an 
error either in the field-notes or in the plot. The bearings of 
all division-fences were taken, as well as their point of inter- 
section with the course, so that these interior lines could be 
plotted and a map of the farm obtained. The " old mill " is 
located by bearings taken from corners B and G. The reverse- 
bearings are given in parenthesis. 



SURVEYING METHODS. 



183 



FIELD-NOTES— COMPASS SURVEY. Oct. 23, 1885. 



No. of 
Course. 



Wt.= I 



Wt.= 1 



Wt.= 3 



4 
Wt.= 



5 
Wt.= 



6 
Wt.= 



Wt.= 3 



Wt. = 1 



Point. 



Bearing-tree... 
Pasture Fence. 
Yard M . 

Orchard " ! 
Corner B 



B. T.... 

Old Mill. 
Fence.. . 



Corner C. 



B. T 

Old Mill.... 

Fence 

Mill Creek. 

Fence 

Corner D... 



B. T .... 
Corner E. 



B. T 

Fence.. . . 
Corner F, 



B. T 

N. bank Mill Creek. 



Corner G. 



Fence 

Offset, 0.40. 

.60. 

.80. 

.70. 

.30. 

" .20. 

Corner H 



Corner A. 



Bearing. 



S. 76 50' 
West.. . . 


E.... 


• « 


« < 


<« 






N. 54° 15 
N. 58 
North.. . . 


E... 
E... 


« < 


S. 89 55' 
(West). . . 


E.... 



N. 22° 20' W. 

N. 26 45' W.. 
N. 6i° 45' W.. 



N. 64 W 

N. 27 40' E. . . 
(S. 27 45' W.) 



S. 85 W 

N. 19 10' W. 
(S. 19° 15' E.). 



S. 62 30' W.. 

South 

N. 86° 50' W. 
(S. 86° 45' E.). 



S. 40 15' E. 



S. 47° 30' W. . , 
(N. 4 7°3o'E.). 



S. 32 E. 



S. 77 3 45'W... 
(N. 77° 45' E.). 



S. 89 W.... 
(N. 89 E.). 



Distance 
along the 
t Course. 



Ch. 

7.20 

9-75 

n-54 

13.90 

25.42 



12.50 
24.10 
34-68 



9.90 
10.70 
12.45 
24.00 



7.40 



15.80 
25-58 



0.30 
0.80 
1.50 



0.00 

0.00 

3.00 

6.00 

9.00 

12.00 

13-60 

13.60 



3-53 



Remarks. 



True bearings given. 
Variation of needle 5 C 

50' east. 
Henry Flagg, 

Compassman. 
PeterLong, 
John Short, 



l\ 



Chainmen, 



Courses 1 and 2 are 
along the centres of 
the highway. 



1 84 



SUR VE YING. 




Due South 
Fig. 54. 



SURVEYING METHODS. 



I8 5 




Fig. 55. 



COMPUTING THE AREA. 

171. The Method stated. — In 
Fig. 55 * let ABCDE be the tract whose 
area is desired. Let us suppose the 
bearings and lengths of the several 
courses have been observed. Pass a 
meridian through the most westerly- 
corner, which in this case is the corner 
A. Let fall perpendiculars upon this 
meridian from the several corners, and 
to those lines drop other perpendicu- 
lars from the. adjacent corners, as shown 
in the figure. Then we have : 



Area ABCDE = bBCDfb - bBAEDfb 

= bBCe + eCDf - (bBA + AEa + aEDf). (1) 

Hence twice the area ABCDE is 



2A = (bB + eC)Bc + (eC+/D)Dd 
- {bB)Ab - (aE)Aa — (aE +/D)Eg. 



(2) 



We will now proceed to show that these distances are all 
readily obtained from the lengths and bearings of the courses. 

172. Latitudes, Departures, and Meridian Distances. — 
The latitude of a course is the length of the orthographic pro- 
jection of that course on the meridian, or it is the length of the 
course into the cosine of its bearing. If the forward bearing 
of the course is northward its latitude is called its northing, and 
is reckoned positively ; while if the course bears southward its 
latitude is called its southing, and is reckoned negatively. 



* The lines OD and OX in this figure are used in art. 185. 



1 86 SURVEYING. 



The departure of a course is the length of its orthographic 
projection on an east and west line, or it is the length of the 
course into the sine of its bearing. If the forward bearing of 
the course is eastward its departure is called its easting, and is 
reckoned positively ; while if its forward bearing is westward 
its departure is called its westing, and is reckoned negatively. 

The meridian distance of a point is its perpendicular dis- 
tance from the reference meridian, which is here taken through 
the most westerly point of the survey. 

The meridian distance of a course is the meridian distance 
of the middle point of that course ; therefore 

The double meridian distance of a course is equal to the sum 
of the meridian distances to the extremities of that course. 
The D. M. D.'s of the two courses adjacent to the reference 
meridian are evidently equal to their respective departures. 
The D. M. D. of any other course is equal to the D. M. D. of 
the preceding course plus the departure of that course plus 
the departure of the course itself, easterly departures being 
counted positively and westerly departures negatively. This 
is evident from Fig. 55. 

Thus in Fig. 55 Dd is the latitude and dC is the departure 
of the course DC. If the survey was made with the tract on 
the left hand, then the latitude of this course is positive and 
the departure negative ; while the reverse holds true if the 
survey was made with the tract on the right hand. In this 
discussion it will be assumed that the survey is made by going 
around to the left, or by keeping the tract on the left hand, 
although this is not essential. The D. M. D. of this course 
CD is fD + eC; or it is the D. M. D. of BC+ cC+(- dC). 

In equation (2), art. 171, the quantities enclosed in brack- 
ets are the double meridian distances of the several courses, 
all of which are positive, while the distances into which these 
are multiplied are the latitudes of the corresponding courses. 
If we go around towards the left the latitudes of the courses 



SURVEYING METHODS. 1 87 

AB, DE, and EA are negative, and therefore the correspond- 
ing products are negative, while the latitudes of the courses 
BC and CD being positive, their products are positive. 

We may therefore say that twice the area of the figure is 
equal to the algebraic sum of the products of the double meridian 
distances of the several courses into the corresponding latitudes, 
north latitudes being reckoned positively and south latitudes 
negatively, and the tract being kept on the left in making the 
survey. If the tract be kept on the right in the survey, then 
the numerical value of the result is the same, but it comes out 
with a negative sign. 

173. Computing the Latitudes and Departures of the 
Courses. — Since the departure of a course is its length into 
the sine, and its latitude its length into the cosine, of its bear- 
ing, these may be computed at once from a table of natural or 
logarithmic sines and cosines. When bearings were (formerly) 
read only to the nearest 15 minutes of arc, tables were used 
giving the latitude and departure for all bearings expressed in 
degrees and quarters for all distances from 1 to 100. Such 
tables are called traverse tables. It is customary now, how- 
ever, to read even the needle-compass closer than the nearest 
15 minutes; and if forward and back readings are taken on all 
courses, and the mean used, these means will seldom be given 
in even quarters of a degree. If the transit or solar compass is 
used, the bearing is read to the nearest minute. The old style 
of traverse table is therefore of little use in modern survey- 
ing. The ordinary five- or six-place logarithmic tables of 
sines and cosines are computed for each minute of arc, and 
these may be used, but they are unnecessarily accurate for or- 
dinary land-surveying. For this purpose a four-place table is 
sufficient. If the average error of the field-work is as much as 
1 in 1000 (and it is usually more than this), then an accuracy 
of 1 in 5000 in the reduction is evidently all-sufficient, and this 
is about the average maximum error in a four-place table; that 



188 



SURVEYING. 



is, the average of the maximum errors that can be made in the 
different parts of the table. 

Table III. is a four-place table of logarithms of numbers 
from I to 10,000, and Table IV. is a similar table of logarithms 
of sines and cosines, from o to 360 degrees. If a transit is 
used in making the survey, and if it is graduated continu- 
ously from o to 360 degrees, then the azimuths of the several 
sides are found, all referred to the true meridian or to the first 
side. If it is desired now to take out the latitudes and de- 
partures, the same as for a compass-survey, where the bearings 



190° 



W90 



/ L " f " 


L+ \ 


/ D — 


D+ \ 


\ L ~" 


L— / 


\ D ~~ 


D+ / 



270 E 



0° 

s 

Fig. 56. 



of the sides are given directly referred to the north and south 
points, it may be done by Table IV. 

Since the log sine changes very fast near zero and the log 
cosine very fast near 90 , the table is made out for every min- 
ute for the first three degrees from these points ; for the rest 
of the quadrant it gives values 10 minutes apart, but with a 
tabular difference for each minute. It is very desirable to 
make the table cover as few pages as possible for convenience 
and rapidity in computation. In this table the zero-point is 



SURVEYING METHODS. 



189 



south and angles increase in the direction SWNE, so that in 
the first quadrarft both latitudes and departures are negative. 
In the second quadrant latitude is positive and departure nega- 
tive, in the third both are positive, and in the fourth latitude 
is negative and departure positive. These relations are shown 
in Fig. 56. For any angle, falling in any quadrant, if reckoned 
from the south point in the direction here shown, the log sin 
(for departure) and log cosine (for latitude) may be at once 
found from Table IV. If these logarithms are both taken out 
at the same time and then the logarithms of the distance from 
Table III., this can be applied to both log sin and log cos, thus 
giving the log departure and log latitude, when from Table III. 
again we may obtain the lat. and dep. of this course, giving 
these their signs according to the quadrant in which the azi- 
muth of the line falls. 

If Table IV. is to be used for bearings of lines as given by a 
needle-compass, then enter the table for the given bearing, in 
the first set of angles, beginning at o and ending at 90 . 

Example: Compute the latitudes and departures of the survey plotted in 
Fig- 55. P- 185, by Tables III. and IV. The following are the field-notes as they 
would appear, first, as read by a transit and referred to the true meridian; and, 
second, as read by a needle-compass: 



Station. 


Azimuth referred to 
the South Point. 


Compass bearing. 


Distance. 


A 


29O 45' 


S. 69° 15' E. 


7.06 


B 


217 15' 


N. 37° 15' E. 


5-93 


C 


140° 30' 


N. 39 c 30' W. 


6.00 


D 


57° 45' 


S. 57° 45' W. 


4-65 


E 


30 00' 


S. 30 00' w. 


4.98 



190 



SURVEYING. 



The following is a convenient form for computing the lati- 
tudes and departures : 



Course 

AB 

4*h Q- 


Course 

BC 

3dQ. 


Course 

CD 

2dQ. 


Course 

DE 

istQ. 


Course 

EA 

istQ. 


9.9708 


9.7820 


9- 8 °35 


9.9272 


9 . 6990 


.8488 


•7731 


.7782 


.6675 


.6972 


.8196 


.5551 


•5817 


•5947 


• 39 62 


+ 6.60 


+ 3-59 


-3-82 


-3-93 


-2.49 


9-5494 


9 . 9009 


9.8874 


9.7272 


9-9375 


.8488 


•7731 


.7782 


.6675 


.6972 


.3982 


.6740 


.6656 


• 3947 


•6347 


— 2.50 


+ 4.72 


4-4.63 


— 2.48 


-4-31 



log sin (dep.) - 
log dist. = 

log dep. = 
Departure '= 

log cos (lat.) = 
log dist. = 

log lat. = 
Latitude = 



It is seen that Table IV. answers equally well for either set 
of bearings, and also that Table III. would have given the lati- 
tudes and departures to the fourth significant figure as well as 
to the third. If the prorier quadrant is given for each course 
in the heading as shown above, then the signs may be at once 
given to the corresponding latitudes and departures. 

174. Balancing the Survey. — If the bearings and lengths 
of all the courses had been accurately* determined, the survey 
would " close ;" that is, when the courses are plotted succes- 
sively to any scale the end of the last course would coincide 
on the plot with the beginning of the first one. Furthermore, 
the sum of the northings (plus latitudes) would exactly equal 
the sum of the southings (minus latitudes), and the sum of the 

* The error of closure simply shows a want of uniformity of measurement, 
for if all the sides were in error by the same relative amount, the survey would 
close just the same. For instance, if an erroneous length of chain were used, 
the survey might close but the area be considerably in error. See arts. 175 

and 177. 



SURVEYING METHODS. 



191 



eastings (plus departures) would exactly equal the sum of the 
westings (minus departures). It.is evident that such exactness 
is not attainable in .practice, and that neither the north and 
south latitudes nor the east and west departures will exactly 
balance, there always being a small residual in each case. 
These residuals are called the errors of latitude and departure 
respectively. The distribution of these errors is called bal- 
ancing the survey. 

In the form for reduction of the field-notes given below, 
wherein this example is solved, it is seen that the error of lati- 
tude is 6 links and the error of departure is 5 links. The dis- 
tribution of these errors is made by one of the following : 



FORM FOR COMPUTING AREAS FROM BEARINGS AND DISTANCES 

OF THE SIDES. 



Sta- 


Courses. 


Dif. Lat. 


Departure. 


Balanced. 


Q 

a 


+ 

Area. 




tions. 


Bearings. 


Dist. 


N. 
•f 


S. 


E. 

+ 


W. 


Lat. 


Dep. 


Area. 


A 


S. 69 15' E. 


Ch. 

7.06 




2.50 


6.60 




— 2.52 


+ 6.61 


6.61 




16.66 


B 


N. 37 15' E. 


5-93 


4.72 





3-59 




+ 4-71 


+ 3.60 


16.8s 


79.22 


.... 


C 


N. 39 30' W. 


6.00 


4-63 







3.82 


+ 4.62 


-3-8i 


16.61 76.74 


— 


D 


S.57°45'W. 


4-6 5 





2.48 




3-93 


— 2.49 


— 3-9 2 


8.88 


.... 


22.11 


E 


S. 30 oo' W. 


4.98 




4-31 





2.49 


— 4-3 2 


— 2.48 


2.48 


.... 


10.71 






28.62 


9-35 


9.29 


10.19 


10.24 




I55-96 


49.48 




1 
Error in lat. 


9.29 


Error in dep 


10.19 
= -05. 




2\2 


49.48 






= .06 


106.48 





Error of closure = 5 "*" = 0.0027 
2862 

sb x in 366. 



Area = 53.24 sq. ch. 
ss 5. 324 Acres 



192 SURVEYING. 



RULES FOR BALANCING A SURVEY. 

RULE i. As the sum of all the distances is to each particular 
distance, so is the whole error in latitude (or departure) to the cor- 
rection of the corresponding latitude (or departure), each correc- 
tion being so applied as to diminish the whole error in each 
case. 

RULE 2. Determine the relative difficulties to accurate 
measurement and alignment of the several courses, selecting 
one course as the standard of reference. Thus, if the standard 
course would probably give rise to an error of 1, determine 
what the errors for an equal distance on the other courses 
would probably be, as ij, 2, 1,0.5 etc « Multiply the length 
of each course by its number, or weight, as thus obtained. 
Then we would have : 

As the sum of all the multiplied lengths is to each multiplied 
length, so is the whole error in latitude (or departure) to the cor- 
rection of the corresponding latitude (or departure), each correc- 
tion being so applied as to diminish the whole error in each 
case. 

These two rules are based on the assumption that the error 
of closure is as much due to erroneous bearings as to erroneous 
chaining,* which experience shows to be true in needle-compass 
work. 

If, however, the bearings are all taken from a solar compass 
(or attachment) in good adjustment, or if the exterior lines are 
run as a traverse with a transit, so that the angles of the pe- 
rimeter are accurately measured, then the above assumption 
does not hold, as it is highly probable that the error of closure 
is almost wholly due to erroneous chaining. Especially would 
this be highly probable if the azimuth is checked by occupying 

* Let the student prove the correctness of rules I and 3 for the assumed 
sources of error. 



SURVEYING METHODS. 1 93 

the first station on closing and redetermining the azimuth of the 
first course, as found from the traverse, and comparing it with 
the initial (true or assumed) azimuth of this course. If it thus 
appears that the traverse is practically correct as to angular 
measurements, it may be fairly assumed that the error of 
closure is almost wholly due to erroneous chaining. In this 
case use 

RULE 3. As the arithmetical sum of all the latitudes is to any 
one latitude, so is the whole error in latitude to the correction to 
the corresponding latitude, each correction being so applied as 
to diminish the whole error in each case. Proceed similarly 
with the departures.* 

In the solution given on p. 191 the first rule is applied. In 
ordinary farm-surveying it is not common to give the lengths 
of the courses nearer than the nearest even link or hundredth 
of a chain. In balancing, therefore, the same rule may be 
observed. 

175. The Error of Closure is the ratio to the whole pe- 
rimeter of the length of the line joining the initial and final 
points, as found from the field-notes. The length of this line 
is the hypotenuse of a right triangle of which the errors in 
latitude and departure are the two sides. Its length is there- 
fore equal to the square root of the sum of the squares of 
these two errors. This divided by the whole perimeter gives 
the error of closure, which ratio is usually expressed by a 
vulgar fraction whose numerator is one, being -g^-g- in the 
above example. 

The error of closure for ordinary rolling country should not 



* It is evident that the courses could here be weighted for different degrees 
of difficulty in the chaining ; but instead of multiplying the lengths of the 
courses by their weights, multiply the latitudes and departures by the weights 
of the corresponding courses, and then distribute the errors in latitude and 
departure by these multiplied latitudes and departures. 
13 



194 SURVEYING. 



be more than I in 300. In city work it should be less than I 
in 1000, and should average less than 1 in 5000. 

176. The Form of Reduction. — On p. 191, the ordinary 
form of reduction is shown. Here the courses are not weight- 
ed for different degrees of difficulty in chaining ; and since it 
was a compass-survey the effect of erroneous bearings is sup- 
posed to equal that from erroneous chaining, and so the first 
rule for balancing is used. The balanced latitudes and de- 
partures having been found, the double meridian distances are 
next taken out. In taking out these it is preferable to begin 
with the most westerly corner, whether this be the first course 
recorded or not. In the example solved on p. 191, it is the 
first corner occupied, but in that given on p. 198 it is not the 
first course. By beginning with the most westerly corner 
(which is equivalent to passing the reference meridian through 
that corner), all the double meridian distances will be positive ; 
otherwise some of them may be negative. If attention be 
paid to signs we may begin at any corner to compute the 
double meridian distances. 

A check on the computation of the D. M. D.'s is that, when 
computed continuously in either direction and from any cor- 
ner, the numerical value of the D. M. D. of the last course 
must equal its departure. This is a very important check and 
must not be neglected, as it proves the accuracy of all the D. 
M. D.'s. 

We are now able to compute the double-areas according to 
equation (2), art. 171, since the terms entering in that equation 
have their numerical values determined. The several products, 
being the partial double-areas, are written in the last two coL 
umns, careful attention being paid to the signs of these prod- 
ucts. Thus, when the reference meridian is taken through the 
most westerly corner, then all the D. M. D.'s are positive and 
the results take the sign of the corresponding latitude. If 
some of the D. M. D.'s are negative, then the signs of these par- 



SURVEYING METHODS. 1 95 

tial areas are opposite to those of the corresponding latitudes. 
The algebraic sum of the partial double-areas is twice the 
area of the figure, as shown in eq. (2), art. 171. If the dis- 
tances are given in chains, then the area is given in sq. 
chains, and dividing by ten gives the area in acres. If the dis- 
tances were given in feet, as it often is, being measured by a 
100-foot chain or tape, then the area is in sq. feet, and this 
must be divided by 43560, the number of sq. feet in one acre, 
to give the area in acres. This is best done by logarithms, as 
shown in the example solved on p. 198. It is preferable to ex- 
press areas in acres and decimals rather than in roods and 
perches, as was formerly the custom. 

On the following page is the reduction of the field-notes 
given on p. 183. Here the several courses have been weighted 
for various degrees of difficulty in the chaining. Thus, the first 
and second courses were along the public highway and on even 
ground. These are taken as the standard and given the 
weight unity. The third course is on very uneven ground and 
is judged to give rise to about three times the error of courses 
one and two per unit's distance. It is therefore weighted 
three. The proper weight to give to the several courses is 
thus seen to depend on the character of the obstructions to ac- 
curate work, and represents simply the judgment of the sur- 
veyor as to the probable relation of these sources of error. 
The short course FG was very difficult to measure, as there 
were precipitous bluffs, and the course GH was also on very 
uneven ground. 

Following the column of weights in the tabular reduction 
are the multiplied distances ; the errors of latitude and depart- 
ure are distributed according to the results in this column by 
Rule Two, p. 192. This survey was also made with a needle- 
compass. 

In the following example the transit was used, and the 



196 



SURVEYING. 









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SURVEYING METHODS. 



l 97 



survey began at A. The azimuth of the line AB (Fig. 57} 

was found by a solar attachment, 

and then the other courses ran as 

a traverse, the horizontal limb of 

the transit being oriented by the 

back azimuth of the last course. 

The azimuths of the courses are 

all referred to the south point as 

zero, and increase in the direction 

SWNE. After the last course 

FA was run, the instrument was 

carried to A and oriented by a 

back sight on F and the azimuth 

of AB again determined. This 

agreed so well with the original 

azimuth of this course that the 

azimuths of all the courses were 

proved to be correct.f 

The error of closure is therefore due to the chaining alone. 
A hundred-foot chain was used so that the distances are all 
given in feet. The obstructions to chaining were about uni- 
form, so the courses are all given equal weight. In balancing, 
Rule Three must be used, since the errors are supposed . to 
come only from the chaining. 

If the errors in latitude and departure had been distributed 
by Rule One, or in proportion to the lengths of the courses, 
the resulting area would have been 56.41 acres, a- difference of 
O.07 acres, or about one eight-hundredth of the total area. 

177. Area Correction due to Erroneous Length of 




Fig. 57.* 



*The lines MB and 00' in this figure are used in art. 186. 

f From the azimuth check here obtained, as compared to the errors in lat- 
itude and departure, decide whether the latter are due mostly to the chaining 
or whether the errors in azimuth have had an equal influence, and so determine 
whether to use rule 1 or rule 3 in balancing. 



198 



SURVEYING. 













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SURVEYING METHODS. 1 99 

Chain. — If the measuring unit has not the length assigned to 
it in the computation, then the computed area will be errone- 
ous. Such an error will not show in the balancing of the work 
or elsewhere, and hence an independent correction must be ap- 
plied for this error. If the chain was too long by one one- 
thousandth part of its length, for instance, then all the courses 
are too short in the same ratio. And since similar plane fig- 
ures are to each other as the squares of their like parts, we 
would have 

true area : computed area :: (iooi) 2 : (iooo) a , 

or true area — yJ-§-|- computed area (nearly) ;* 

or, in general, if / = length of chain and Al = error in length, 
being positive for chain long and negative for chain short, and 
if Al is small as compared with /, as it always is in this case, 
then if we let 

A = true area, A' == computed area 
C A = correction to computed area, 
and A = relative error of chain, 

1+2AI 
we have A = -—, A' = (1 -f- 2A)A' ; 

whence, A — A' = C A = 2 A A'. 

That is to say, the relative area correction due to erroneous 
length of chain is twice the relative error of the chain, being 
positive for chain long, and negative for chain short. 

* The error in this approximation is one one-millionth in this case, and 
would always be inconsiderable in this class of problems. 



200 SUR VE YING. 



FINDING THE AREA OR SUPERFICIAL CONTENTS OF LAND 
WHEN THE RECTANGULAR COORDINATES OF THE COR- 
NERS ARE GIVEN WITH RESPECT TO ANY POINT AS AN 
ORIGIN. 

178. Conditions of Application of this Method.— 

Where many tracts of land, all bounded by straight lines, are 
somewhat confusedly intermingled, as is the case in many of 
the older States, and where the area of each tract over an ex- 
tended territory is to be found, this method is greatly to be 
preferred to that by means of the boundary-lines. In this case 
it is only necessary to make a general coordinate survey of the 
whole territory, as described in Chapter VIII., on Topographi- 
cal Surveying, using the stadia for obtaining distances, and be- 
ing careful to locate every corner of each tract. If areas alone 
are required, no attention need be paid to the obtaining of 
elevations for contour lines, and so the work is greatly facilitated. 
A transit and two or three stadia rods would be the instru- 
ments used. The survey would then be carefully plotted and 
the coordinates measured on the sheet, or they could be com- 
puted from the field-notes. If the plotting is carefully done 
the former method is preferable. It is best to choose the 
origin of coordinates entirely outside the tract and so that the 
whole area falls in one quadrant, thus making all the coor- 
dinates of one sign. 

Large tracts of mineral land are sometimes acquired by 
large companies, including perhaps hundreds of individual es- 
tates. In such cases a topographical map of the region is 
necessary; and when this survey is made, a little extra care to 
obtain all the " corners" of private claims will enable the areas 
of all such lots to be determined with great accuracy and at 
small additional cost. The method probably has no advan- 
tages when the area of but a single tract is desired. 



SURVEYING METHODS. 



201 



179. The Method of Finding the Area from the Rec- 
tangular Coordinates of the Corners is as follows : 

Let Fig. 58 be the same tract as that given in Fig. 55, and 




let the origin be one chain west of A and three chains south of 
B. Then, from the balanced latitudes and departures for this 
case, given on p. 191, we find the following coordinates of the 
corners y a} y b , etc., denoting the latitudes of the corners A, B, 
etc., and similarly with x at x bi etc., for departures : 



y a = 5-52, j & = 3-oo, #, = 7.71, y d = 12.33, ^ = 9.84. 
x a =i.oo, x b = ?.6i, x a =11.21, ^ = 7.40, ^ = 3.48. 

The area of the figure ABCDE is equal to the areas 



y b BCy c + y c CDy d -r- \y e EDy d + y a AEy e + y b BAy a \ ; 



202 



SURVEYING. 



or 



A = i [(jro—fb) 0&+O + (fa-yc) (*o + *d)-(yd -y e ) {x d +x e ) 

- Oe - ya) (*e + *a) ~ ( Ja ~ ?b) (*« + *&)]• (0 

By developing equation (i) we obtain 
A—\ [y a x e — y a x h +y b x a — y h x c +y c x b — y c x d 

+ yd* — yd** + y e *d — y**^ (2) 

From this we may obtain either of the following: 



^ = i [><* (*• — *b) +yb {?* — *o) +y c O& — x d ) 

+y d (*o — *e) +y e fa — *a)] ; 

or 

A = — i[x a (y e —y b ) + x b (y a -y c ) + x (y b -y d ) 

+ *d (y c -ye) + *e (y d - y a )l 

From these equations we may obtain the following 



(3) 



RULE FOR FINDING THE AREA OF A CLOSED FIGURE 
BOUNDED BY STRAIGHT LINES FROM THE RECTANGULAR 
COORDINATES OF THE CORNERS. 

Multiply the \ a f? s ^ [ to each corner by the difference be- 
tween the \ 1 • !■ of the two adjacent corners, always making 

the subtraction in the same direction around the figure, and take 
half the sum of the products. 

The student will observe that this is simply a more general 
case of the former method of computing the area from the 
latitudes and double-meridian distances. 



SURVEYING METHODS. 



203 



180. The Form of Reduction for this case is given below. 



Corner. 


Ordinates 

6# 


Abscissae 


Difference 
between Alter- 
nate Abscissae. 


Double Areas. 


A 
B 
C 
D 

E 


5.52 
3.00 

7.71 

12.33 

9.84 


1.00 
7.61 

II. 21 

7.4O 
3-48 


- 4-13 

— I0.2I 

+ .21 

+ 7-73 
+ 6.40 


— 22.8o 

— 30.63 
-{- I.62 

+ 95-31 
-f 62.98 



Plus areas = 159.91 
Minus areas = 53.43 

2 ) 106.48 
Area = 53.24 sq.chns. 
= 5 . 324 acres. 



This is the same result as found on p. 191 by the other 
method, as it should be, since the same balanced latitudes and 
departures were used in each case. 

It is also evident that after the balanced latitudes and 
departures are obtained for the ordinary perimeter-survey, the 
area may be computed by this form — from equations (3), p. 
202, if preferred. Or, if the coordinates of the corners are 
taken at once from a map, or computed from traverse lines, 
the bearings and lengths of the courses joining such corners 
could readily be computed. Thus, the length of any course, 

as BC, is BC = V{x c — x^f -\- (y c — y b f t while its bearing is 
the arc whose tan is — . 

y c — y* 
181. Supplying Missing or Erroneous Data. — In any 

closed survey there are two geometric conditions that must 
be fulfilled, viz. : 

1. The sum of all the latitudes must be zero. 

2. The sum of all the departures must be zero. 



204 SURVEYING. 



These two conditions give rise to two corresponding equa- 
tions. 

If /,, / 2 , /„ etc., be the lengths of the several courses, and if 
#i> a > ^» etc «> be their compass-bearings, then our two geo- 
metric conditions give 

/ x sin 6^ + ^ sin a + /, sin 0, + etc., ==o. . . (i) 

l x cos 0, 4" K cos 2 4" h cos 3 -)- etc., = o. . . (2) 

Since we have two independent equations, we can solve for 
two unknown quantities. These two unknowns may be any 
two of the functions entering in the above equations. Thus, 
if any two distances, any two bearings, or any one distance 
and any one bearing are missing, they may be found from 
these equations. Or, if but one bearing or distance is missing, 
it may be found from one of these equations and the other 
equation used for balancing either the latitudes or departures. 
When all bearings and distances are given, these equations are 
really used in balancing ; but if they are both used to deter- 
mine missing quantities, there can be no balancing of errors, 
for when the missing quantities are computed by these equa- 
tions, both latitudes and departures will exactly balance. In 
other words, all the errors of the survey are thus thrown into 
these two quantities. 

This artifice should therefore never be resorted to except 
where it is impracticable to actually measure the quantities 
themselves in the field. 

There are four cases to be solved : 

I. Where the bearing and length of one course are un- 
known. 

II. Where the bearing of one course and length of another 
are unknown. 

III. Where the bearings are unknown. 

IV. Where two lengths are unknown. 



SURVEYING METHODS. 205 

The bearings will be reckoned from both north and south 
points around to the east and west points, as is common in 
compass surveying. Then the length of a course into the sin 
of its bearing gives its departure, and into the cos of its bear- 
ing gives its latitude. North latitude is plus and south latitude 
minus; east departure plus and west departure minus. 

In every case let the sum of the departures of all known 
courses, taken with the opposite sign, be D, and the sum of 
their latitudes, taken with the opposite sign, be L. Then D 
and L are the departure and latitude necessary to close the 
survey. 

Case I. — Bearing and length of one course unknown. 

The two condition equations here become 

4 sin 6 m = D ; ) 

l m cosd m = L. J {3) 

D 
Whence tan 6 m = -y (4) 

Having found the bearing, find l m from either of equations 
(3). Particular attention must always be paid to the signs of 
D and L. Evidently sin 6 m (dep.) and cos 6 m (lat.) have the 
same signs as D and L respectively, whence the quadrant 
which includes the bearing may be determined and the proper 
letters applied. For this purpose Fig. 56 may be consulted. 

Case II. — The bearing of one course and the length of another 
unknown. 

In this case let a be the known bearing of the course whose 
length is unknown, and let / be the known length of the course 
whose bearing is unknown. Then we have 

/ w sin* + /sin 6 n = D-,\ 
l m cos a-\- 1 cos 6 n = Z. f 



206 SUR VE YING. 



If we let sin a = s, and cos a = c, we have 



l m z=sD + c/± Vr - (D 2 + U) + (sD + cL)\ . (6) 

Here there are two values of l m which will satisfy the equa- 
tion, and so there are two solutions to the problem. If the 
surveyor has no knowledge whatever of either the unknown 
length or bearing, the problem is indeterminate. If he has 
seen the tract he could usually tell which length or which 
resulting bearing was the correct one, when the problem would 
become determinate. When l m is found, substitute in one of 
equations (5) and find 6 n . Pay careful attention to the signs 
of the trigonometrical functions of all bearings. 

CASE III. — When two bearings are unknown. 

Let I' and /" be the known lengths of the courses whose 
bearings are unknown. Then the equations become 



/'sin^ + Z" sin# n = £>; 
I' cos6 m + I" cos 6 n = L. 



Whence cos o n = ; . . . (8) 



/"* _ /« _i_ d* 4. u 
Where K = -^r— 1 — 



This case is also indeterminate unless one is able to tell 
which of the two sets of bearings is the correct one. 

Case IV. — When the lengths of two courses are unknown. 

Let a and b be the known bearing of the courses whose 
lengths are unknown. 



SURVEYING METHODS. 207 



Our equations here become 



l m sin a -f- 4 sin b = D ; 
/ m cos # -f- / n cos b = L. 



(9) 



_ Z? cos # — L sin # 

whence 4 = : — ? T\ • X ( I0 ) 

sin (a — b) 



This case is determinate. 

In case there is but one unknown, then either one of equa- 
tions (3) will solve. In taking out either the sine or the cosine 
from the tables, however, two angles will always be found 
equidistant from the east or west point if the sine, and equi-' 
distant from either the north or south point if the cosine, 
either of which may be chosen. In such case both sine and 
cosine must be found, when the signs alone of these two func- 
tions will determine the quadrant in which the bearing is found. 
Hence, if the single unknown is a bearing ; both of the equa- 
tions (3) must be used in order to determine which of the two 
bearings given by the table is the correct one, but one alone is 
sufficient to obtain the numerical value of the bearing. Thus, 
if the sine equation is used to compute the bearing, then the 
latitude may be taken out for the given length and bearing; 
and these will then not balance, but will have to be balanced 
in the usual way, while the departures will, of course, balance, 
since the residual departure D necessary to close the survey as 
to departures was used to compute the corresponding bearing. 
The reverse of this would be true if the cosine equation were 
used to compute the bearing. 



208 SURVEYING. 



PLOTTING THE FIELD-NOTES. 



181. To plot a Compass Survey select a point for the 
initial station, and pass a meridian through it in pencil. By 
means of a semicircular protractor, such as is shown in Fig. 
44, mark the bearing and draw an indefinite line from the sta- 
tion point. On this line lay off to scale the length of the 
course, thus establishing the next corner. Through this draw 
another pencil meridian, and proceed as before. If the plot- 
ting is perfect the length of the line joining the final with the 
initial point, taken to scale, is the error of closure of the sur- 
vey ; and the horizontal and vertical components of this line, 
taken to scale, should be the errors in departure and latitude 
respectively as obtained by the computation. 

If preferred, the bearings of the successive courses may be 
so combined as to give the deflection-angle at each station, and 
these laid off from the preceding course as already drawn. 
Errors are more likely to accumulate in the plot by this 
method, however, than by that first given. 

Again, the rectangular coordinates of the several corners 
may be computed and these plotted from a pair of rectangular 
axes, but this is not a common practice. 

For the plotting of transit surveys, especially where the 
stadia is used, see Chapter VIII. 



THE AREAS OF FIGURES BOUNDED BY CURVED OR IRREGULAR 

LINES. 

182. The Method by Offsets at Irregular Intervals. 

— Where a tract of land is bounded by a body of water, as a 
stream or lake, it is customary to run straight lines as near the 
boundary as practicable and then to take rectangular offsets 
at selected intervals from these bordering-lines to the irregular 
boundary. These small areas are then computed as trapezoids, 



SURVEYING METHODS. 



209 




Fig. 59. 



the distance along the base-line being the altitude and the half- 
sum of the adjacent offsets being the mean width. The offsets 
should therefore be run at such intervals as to make this 
method of computation sufficiently accurate. Such offsets 
were taken from the course GH in Fig. 54, the notes for which 
are given on p. 183. 

The work of computation may be shortened by using a 
modified form of the method 
of areas from the rectangular 
coordinates of the corners, 
which, in this case, are the ends 
of the offset lines. Let Fig. 
59 be an area to be determined 
from the offsets from the line 
AK. The position and lengh of the offsets are given. Take 
the origin at A and let the distances along AK be the abscissae, 
and the lengths of the offsets be the ordinates. Using the 
second of equations (3), p. 202, we have 

A = *i [x a {y k — y b ) + x b (y a —y c ) + x e (y b —y d ) 

+ Xd(y e - y e ) + *e (y d -y f ) 
+ Xf(ye-y g ) + * g (y f -n) 

+ *h (y g - n) + *ic (y h - y a )\ ) 

But here x ay x b > y af and y k are all zero ; also x h = x k , hence 
this equation becomes 

A = i \x c (y h -y d ) -f x d (y e - y e ) + x e {y d - y f ) 

+*f(ye-y g )+x g (y/-y h )+ x h (y g +y h )l(2) 



From eq. (2) we have the 



* The plus sign is here used, since we have gone around the figure in a direc- 
tion opposite to that followed in the general case 



2IO 



SURVEYING. 



RULE FOR FINDING AREAS FROM RECTANGULAR OFFSETS AT 

IRREGULAR INTERVALS. 

Multiply the distance along the course of each intermediate 
offset from the first by the difference between the two adjacent 
offsets, always subtracting the following from the preceding. 
Also multiply the distance of the last offset from the first by the 
sum of the last two offsets. Divide the sum of these products by 
two. 

The following is the numerical reduction for finding the 
area of the irregular tract shown in Fig. 59. 



Offset. 


Distance 
from A. 


Length of 
Offset. 


Differences. 


Products. 




ch. 


ch. 


ch. 


sq. ch. 


B 


O.OO 


1-53 






C 


1. 21 


I.76 


- O.47 


— O.57 


D 


2.23 


2.00 


— O.56 


— I.25 


E 


3-56 


2.32 


+ -09 


+ .32 


F 


5-04 


1. 91 


+ -87 


+ 4.38 


G 


5.75 


1-45 


+ -9i 


+ 5.23 


H 


7.00 


1. 00 


+ s-45 


+17.15 



2 ) 25.26 



Area = 12.63 sq. chs. 
= 1 . 263 acres. 

It is evident that an area bounded on all sides by irregular 
or curved lines could have a base-line run through it, and off- 
sets taken from this line to both boundaries and the area com- 
puted by this method. Example 196, p. 221, should be so 
computed. 

183. The Method by Offsets at Regular Intervals. — If 
the intervals between the offsets, or ordinates, are all equal the 
computation is much simplified. On the assumption that the 
area is a series of trapezoids, we have the 



SURVEYING METHODS. 211 



RULE FOR FINDING THE AREA FROM RECTANGULAR OFFSETS 
AT REGULAR INTERVALS. 

Add together all the intermediate offsets and one half the end 
offsets, and multiply the sum by the constant interval between 
them. 

The following rules for finding areas are found from the suc- 
cessive orders of differences in each case and may all be derived 
by a rigid development.* They assume that the bounding-line 
is curved and that rectangular ordinates have been measured 
at uniform intervals from a base-line traversing the figure. 

Let the common interval between ordinates be d; let the 
lengths of the ordianates be h , h v h 2 . . . . h n ; and let the 
number of intervals be N. 

L N =a I, A = - (h -f /*,), Trapezoidal Rule. 

*4 



II. N = 2, A — - (h -f- 4^ -f-^ 2 ), Simpson's i Rule. 

III. N = 3) A = ^(K+ iK + IK + K), Simpson's f Rule. 

2d 
\V.N= A , A = ^7 (*. + *0 + 32 (*, + A.) + ia«. 

V. N = 6, ^ = ]£ [A. + ^ 2 + <&,+ /&,+ 5 (K+ A.+ *.)+*,]. 



This is called Weddel's Rule. If a quadrant be computed by 

this rule, the result is 0.779^ instead of 0.785^, the true value. 

When an area, bounded by a base line and two end ordi- 

*See appendix C. 



212 



SURVEYING. 



nates, be divided by imaginary lines parallel to the end ordi- 
nates and equally spaced, as in Fig. 60, and if the middle ordi- 




nates of these partial areas be measured, then if d = common 
width of the partial areas and k x , h„ k 3 , etc., their middle ordi- 
nates, a the first end ordinate and b the last one, we have, 
approximately, 

I. A = d2h, 



where 2/i signifies the summation of all the Its. 
The following rules are, however, more accurate : 



II. A = d^h -\- — {a — h x -\-b — h n \ Poncelet's Rule ; 



or, 



[Rule. 



III. A = d2k + — (Sa+/i,-9k l +8Z> + /in-i-9kn), Francke'-s 
72 



The various rules above given are often used to determine 
areas of irregular figures such as steam diagrams, cross-sections 
of structural forms, streams, excavations, etc. The most 
ready and accurate means of determining all such areas, how- 
ever, is by means of the planimeter. 



SURVEYING METHODS. 



213 



THE SUBDIVISION OF LAND. 

184. The Problems arising in the subdivision of land are 
of almost infinite variety. All such problems are solved by the 
application of the fundamental principles and relations of 
geometry and trigonometry with which the student is supposed 
to be familiar. There are, however, two classes of problems of 
such frequent application that they will be given in detail. 

185. To cut off from a Given Tract of Land a Given 
Area by a Right Line, starting from a Given Point in the 
Boundary. — In Fig. 55, p. 185, let O be the middle point on 
the line AB, from which a line is to be run in such a manner 
as to cut off three acres from the western portion of the tract. 
We may at once assume that the dividing-line will cut the side 
DC in some point Jf, whose distance from D is to be found. 
First compute the area OAED, using the balanced latitudes 
and departures given on p. 191, we have the following : 



Course. 


Lat. 


Dep. 


D. M. D. 


Double Areas. 


+ • 


- 


AO 
OD 
DE 


ch. 
— I.26 
f£ 8.07) 

jar 2.49 

W- 4.32 


ch. 

+ 3-30 

(+ 3.IO) 

- 3.92 

— 2.48 


3-30 
9.70 
8.88 
2.48 




4.16 


78.28 


22.11 

IO.71 







(- 8.07) (- 3.10) 



Sums + 78.28 
- 36.98 



36.98 



2 ) 41.30 

Area = 20.65 sq. chs. 
= 2.065 acres. 



Here the latitude and departure of the course OD are such 
as to make the latitudes and departures balance. The area is 



214 SURVEYING. 



found to be 2.065 acres, leaving 0.935 acres to be laid off from 
OD by the line OX. It remains now to find the point X, 

First compute the length and bearing of the line OD from 
Case I., p. 205. 

Thus we have 

tan 6=y- = i-i— = 0.384. 
L -f- 8.07 D * 

Whence 6 = 21 from the table of natural tangents. From 
the table of natural sines, we find sin 21° = 0.358. 
Hence from eq. (3), p. 205, we have 

/sin 6 = Z>, or 0.358/= 3.10. 

Whence /= 8.66 chains. 

The bearing is evidently N. 21 E. 

We now have to find the distance DX such that the area 
ODX shall be 9.35 sq. chains. Since the area of any triangle 
is one half the product of two sides into the sine of the in- 
cluded angle (another way of saying it is equal to half the base 
into the altitude), we have 

9.35 = 1(8.66 X DX) sin ODX. . . . . (1) 

From the bearings of OD and DX we find the angle ODX 
to be 6o° 30', hence sin ODX = 0.870, from which we find 

DX— 2.48 chains. 

The length and bearing of the line may OX be computed 
from its latitude and departure, the same as was done for the 
line OD above, or we may compute the angle DOX and length 



SURVEYING METHODS. 2\$ 

OX by solving the triangle DOX. The bearing of OX may 
then be found, and the line run from 0. There will then be 
two checks on the work, viz. : the measured lengths of OX and 
DX must be equal to their computed values. 

To find the angle DOX, let the three angles of the triangle 
be D, O, and X, and the sides opposite these angles be d> o, 
and x t respectively. Then we have 

tan i (X - 0) = J^ tan f (X+ O) ; 
whence O = | (X+ O) - \ {X - O). 

Also, ,-OX^iOD-BX)^^. 

We therefore have the following 

RULE FOR CUTTING OFF A GIVEN AREA BY A LINE START- 
ING FROM A GIVEN POINT IN THE BOUNDARY 

Having first surveyed the tract and plotted the same, join 
the given point on the plot with the corner which will give the 
nearest approximation to the desired area. Compute the 
length and bearing of this line, and of the area thus cut off. 
Subtract this area from the desired area, and the remainder is 
the area to be cut off in the form of a triangle, one side of 
which has bearing and distance given, and another side has its 
bearing alone given. From these data compute the lengths and 
bearings of the other sides, one of which is the line sought. 
This line may then be run, and its length measured, as well as 
the length of the portion of the opposite boundary cut off, for 
a check on the accuracy of the work. 

186. To cut off from a Given Tract of Land a Given 
Area by a Right Line running in a Given Direction. 



2l6 



SURVEYING. 



— Let the problem be to cut off 30 acres from the northern 
portion of the tract shown in Fig. 57, p. 197, by a line whose 
bearing is N. 8o° E., or whose azimuth is 260 .* 

Pass a line parallel to the required line through the corner 
nearest to the probable position of the desired line. Let MB, 
Fig. 57, be such a line. Compute the lengths of the lines EM 
and MB by Case IV., p. 206. 

From the computation, p. 198, we have the following: 



Courses 


Azimuth. 


Lengths. 


Balanced 
Latitudes. 


Balanced 
Departures. 


D.M. D.'s 


Double Areas. 


BC 
CD 
DE 

EM 
MB 


205° 39' 
112 12 

55 00 

04 

260 OO 


IOO4 ft. 
896 
912 


-|- 906 ft. 

+ 339 
— 522 


+ 432 ft. 

- 834 

- 750 


2738 
2336 

752 

I 
"53 


+ 2,480,628 
+ 79 I »8o4 

- 392,544 

— 926 
+ 234,059 


(926) 
(II7I) 


— 926 

+ 203 


— I 

+"53 



(+.723) (-1152) 



2 ) 3,II3,02I 



Therefore to close requires L = — 723 and D — -\- 1152. Area = 1,556,510 

sq. ft. 
= 35-73 ac's. 

From equation (10), p. 207, we have 



EM = 



D cos 260 — L sin 260 
sin 259 56' 

(+ 1 152) (+ .1736) - (- 723) (+ .9848) 
+ .9846 



200 + 712 
.9846 



= 926 ft. 



* In this problem it would have shortened the operation somewhat if the 
meridian of the survey had been taken parallel to the dividing-line. The bear- 
ings could have all been changed to give angles from this meridian, and original 
computation made from these new bearings. 



SURVEYING METHODS. 



2i; 



Whence from eq (9), we have 
D - EM sin 4' 



MB = 



sin 260 

+ 1 152 — (926) (— .001 1) 
+ .9848." 



= 1 171 ft. 



Inserting these values of the lengths of the courses EM 
and MB, we can compute the area BCDEM. This is found to 
be 35.73 acres, or 5.73 acres too much. The problem now is 
to pass a line north of MB and parallel to it, so that the area 
included between the parallel lines and the intercepted por- 
tions of ii.Fand BC shall be 5.73 acres, or 249,710 sq. ft. Let 
00' be such a line. This line can be run when either MO or 
BO' is known. It is best, however, to compute both these 
distances, using one for a check. To find these distances, 

Let x = perpendicular distance between the parallel lines 
MB and 00'. 

Let angle EMB = EOO' = 0, 

and angle OO'B = <p. 

Then we have 



Area MOO'B = MB .x — \x> cot -f \x* cot 



= MB .x + ix* (cot - cot 6). . . (0 

Since 0and are known angles, their cotangents are known 
quantities in any case. So, for simplicity, let 

(cot — cot 6) = K\ 



218 SURVEYING. 



also, let the distance MB — D, 

and area MOO'B = A. 

Then the equation becomes 

A^Ztx + iKx*) (2) 



2 2_D _ 2_A 

x -f- K x - K , 



D 2_A £_ 

*- ~K±y -k+k* 



D i 



= -£±xV2AK+D*; 



= x{±V2AK+D*-D) (3) 

That sign of the radical is to be used which will give a 
positive value to x. The other sign would give the value of 
x to be used in laying off the given area on the opposite side 
of MB, provided the sides OM and O'B were continuous in 
that direction. 

Using equation (3) for the problem in hand, we have 

= 79 56'' ; 

0=54° 21'; 

A = 249,710 sq. ft. ; 

JD= 1171 ft.; 

K= 0.7172 - 0.1775 = o 5397; 



SURVEYING METHODS. 219 

whence x = — — (± ^269,537+ 1,371,241 - 1 171) 
= 203.6 feet. 

We can now find MO and BO' from 



MO = -?—^ and £0' 



sin 0* " sin ' 

whence MO — 206.8 feet, and BO' = 250.6 feet. 
The length of the line 00' is 

00' = MB + x (cot — cot 6). 
We may therefore write the following 

RULE FOR CUTTING OFF A GIVEN AREA BY A LINE PASSING 

IN A GIVEN DIRECTION. 

Having first surveyed the tract and plotted the same, pass 
a line on the plot in the required direction through the corner 
which will give the nearest approximation to the desired area. 
Compute the lengths of the two unknown courses bounding 
this area, and then the area itself. Subtract this from the 
given area, and the remainder is the area which is to be cut off 
by a line parallel to the first trial line. This auxiliary area will 
always be a trapezoid, whose area, the length and bearing of 
one of the parallel sides, and the bearings of the remaining 
sides are known. The lengths of these sides may then be 
computed, one of the end lengths laid off, and the dividing- 
line run. Measure the length of this line and also of the other 
end line for checks. 



220 



SURVEYING. 



EXAMPLES. 

187. Compute the area, plot the survey, and determine error of closure 
from the following field-notes : 



Station. 


Bearing. 


Distance. 


A 


S. 46F E. 


20.00 ch. 


B 


S. 74i E. 


30.95 


C 


N. 33i E. 


18.80 


D 


N. 56 W. 


27.60 


E 


W. 


21.25 


F 


S. 5if W. 


13.80 



Answer] Area = 104.43 acres. 

( Error of closure = 1 in 201. 



This being a compass-survey, the errors in latitude and departure must be 
distributed in proportion to the lengths of the courses, regardless of their bear- 
ings, or according to Rule 1, p. 192. If the errors in the bearings (or deflection 
angles) had been very small as compared with the errors in measuring the dis- 
tances, as is the case when the deflection angles are measured with a transit, 
then Rule 3, p. 193, should have been used. 

This would have changed the result by 0.08 acres, the result then being 
104.35 acres. 

188. Find the area and error of closure from the following field-notes : 



Station. 


Bearing. 


Distance. 


A 


E. 


130 rods. 


B 


N. 8° E. 


137 


C 


N. 81 W. 


186 


D 


S. 


54 


E 


S. 36 W. 


125 


F 


S. 45 E. 


89 


G 


N. 40 E. 


70 



SURVEYING METHODS. 



221 



What would be the resulting difference in area from the use of Rules I and 3 ? 

189. In the example, art. 187, suppose the length and bearing of the first 
course were unknown. Let these be found as in Case I., art. 180. 

190. Suppose the length of course A and bearing of B are unknown in same 
example. Compute by Case II. 

191. Let the first two bearings be unknown. Compute them by Case III. 

192. Let the lengths of the first two courses be unknown. Find them by 
Case IV. 

193. Let it be required to cut off twenty-five acres from the west end of 
the tract given in art. 187 by a line passing through a point on the course BC 
at a distance of ten chains from B. Find the length and bearing of the division- 
line, and the other intersecting point on the boundary. 

194. Let it be required to divide the tract given in art. 187 into three equal 
portions by north and south lines. Find the lengths and points of intersection 
of such lines with the boundary-lines. 

195. Compute the coordinates of the corners of the tract given in art. 187, 
taken with reference to a point 35 chains directly south of A, and then com- 
pute the area of the tract from these coordinates by the formula given in art. 
179. This area should, of course, be the same as that obtained by any other 
method where the same balanced latitudes and departures are used. 

196. An irregular tract of land has a straight line run through it and rec- 
tangular offsets taken to the boundary. Find the area of the tract from the 
following notes : 



Distance. 


Width. 


ch. 


ch. 


O 


2-35 


IO 


8.42 


14 


I2.6o 


20 


H.38 


25 


IO.75 


28 


6.15 


30.50 


O.OO 



Is it significant whether or not this tract lies on both sides or wholly on one 
side of the base-line? 

196a. Compute the area of the tract of which the following are the field- 



222 



SUR VE YING. 



notes. The rectangular offsets are taken on both sides of a straight axial line 
R signifying right and L left. 



Distances. 


Side. 


Width or 

Length of 

Offset. 


Distances. 


Side. 


Width or 

Length of 

Offset. 


ch. 




ch. 


ch. 




ch. 


o 


R 


4.23 


18 


R 


I5.80 


o 


L 


O.OO 


20 


L 


5.00 


5 


R 


7.16 


25 


R 


I2.20 


7.5o 


L 


3-45 


30 


L 


2.62 


10 


R 


12.68 


30 


R 


6.48 


10 


L 


6.00 


30 


L 


0.00 


12 


R 


iq.75 









Note. — For a valuable paper on the Judicial Functions of the Surveyor, by 
Judge Cooley of the Michigan Supreme Court, see Appendix A. 



CHAPTER VIII. 

TOPOGRAPHICAL SURVEYING BY THE TRANSIT AND 

STADIA.* 

197. A Topographical Survey is such a one as gives not 
only the geographical positions of points and objects on the 
surface of the ground, but also furnishes the data from which 
the character of the surface may be delineated with respect to 
the relative elevations or depressions. 

198. There are three general methods of making such a 
survey. 

Firsts with a compass (or transit) and chain, to determine 
geographical position, and with a level for obtaining relative 
elevations. 

Second, with a plane-table, either with or without stadia- 
rods. 

Third, with a transit instrument and stadia rods. 

The first method is very laborious, slow, and expensive. It 
is therefore not adapted to large areas. The second method 
has been more extensively used for this purpose than any 
other. The use of the plane-table is fully described in Chap- 
ter V. This method is giving place, however, to the third, 

*The word "stadia" is Italian and was originally used to designate the 
rod used by the inventor of the method. It is now too firmly established to 
be changed. On the U. S. Coast and Geodetic Survey the word "telemeter" 
is used in place of "stadia," but this, which very properly means distance-meas- 
urer, has been appropriated for other appliances used for measuring at a dis- 
tance, as temperature, for example. It would therefore seem that " stadia" 
is the better word to use. 



224 SURVEYING. 



which has been in use in America since about 1864, when it 
was officially adopted on the United States Lake Survey. 
'The system was first used in Italy about 1820. In what fol- 
lows, the third method will alone be described. 

199. The Principle of the location of points by the transit 
and stadia, both horizontally and vertically, is that of polar 
coordinates. That is, the location of the point geographically 
is by obtaining its angular direction from the meridian through 
the instrument, which is read on the limb of the transit, and 
its distance from the instrument, which is read through the 
telescope on the stadia-rod which is held at the point. This 
distance is found by observing what portion of the image of 
the graduated rod is included between certain cross-hairs in the 
telescope. The farther the rod is from the instrument, the 
greater is the portion of the rod's image which falls between 
the cross-wires. 

For elevation, the vertical angle is read on the vertical circle 
of the transit, when the telescope is directed towards a point 
of the stadia-rod as far from the ground as the telescope is 
above the stake over which it is set. The tangent of this 
angle of elevation, or depression, into the given horizontal dis- 
tance is the amount by which the point is above or below the 
instrument station. 

In this way, both the chain and levelling-instrument are dis- 
pensed with, and the slow and laborious processes of chaining 
over bad ground, and levelling up and down hill, are avoided. 
The horizontal distances are obtained as well, m general, as 
by the chain; and the levelling may be done within a few 
tenths of a foot to the mile which is amply sufficient for topo- 
graphical purposes. 

THEORY OF STADIA MEASUREMENTS. 

200. Fundamental Relations.— In Fig. 61 let LS be any 
lens, or combination of lenses, used for the object-glass of a 
telescope. 



TOPOGRAPHICAL SUP VE YING. 



225 



Let A 2 B 2 be a portion of the object (in this case the stadia- 
rod), and let A X B X be its image. The point of the object A^ has 
its image formed at A v and so with B^ and B x . 

Let F be the position of the image for parallel rays, or for 
an object an infinite distance away ; and let C be the centre of 




Fig. 61. 



the instrument, or the intersection of the plumb-line, extended, 
with the axis of the telescope. 

Let E x and Z? a be the " principal points," * and let the 
distance FE X = f (focal length), 



OB f f ( con J u g ate foci )> 
A X B X = i (for image, intercepted portion), 
AJZ^ = s (for stadia, intercepted portion). 
Then, since A X E X is parallel to A 2 E„ and B X E X is parallel to 
B 3 E„ we have 



or, 



A X B X \AJB, :: IE X : 0E 2 , 
i:s ::/ x :/ a . . . 



(1) 



Also, from the law of lenses we have 



* As optics is generally taught in the English textbooks, E x and E 2 are 
made to coincide in a point at or near the centre of the lens; and this is called 
the "optical centre." The "principal points" of the ordinary objective fall 
inside the surfaces of the lens, but they never coincide. The ordinary theory 
is sufficiently approximate for the development of stadia formulae but it saves 
confusion to make the conditions rigid, and it is equally simple. 
15 



226 SURVEYING. 



f> + A~? (2) 

On these two equations rests the whole theory of stadia 
measurements. 

Since the distance FE X =f= focal distance, is a constant 
for any lens or fixed combination of lenses, we see from equa- 
tion (2) that if the object P approaches the lens the distance 
f t is diminished, and therefore f x must be increased ; that is, 
the image recedes farther from the lens as the object ap- 
proaches it, and vice versa. 

If the extreme wires in the reticule of the telescope be sup- 
posed to be placed at A x and B x in the figure, then A X E X B X is 
the visual angle which is equal to A 2 E 2 B^. But as the image 
changes its distance from the objective as the object is nearer 
to or farther from the instrument, so the reticule is moved 
back and forth,* for it must always be in the plane of the 
image. Therefore IE X = / x is a variable quantity, while A X B X 
is constant for fixed wires. Therefore the visual angles at E x 
and E^ are variable. 

If these angles were constant, the space intercepted on the 
rod, and the distance of the rod from the objective, would be in 
constant ratio. Since this is not true, we must find the rela- 
tion that does exist between the distance Efi and the space 
intercepted on the rod, A 2 B 2 . 

From equation (1) we have 

1 s 

A~¥J 



but from equation (2) - = -> — -. 

/1 J Ji 



* If the objective is moved in focusing it does not appreciably affect these 
relations. 



TOPOGRAPHICAL SURVEYING. 227 



Equating these two values of -p, we have 

J\ 

s I I 

or 



/. = {* + A (3) 



that is, the distance of the rod from the objective is equal to 
the intercepted space in the rod multiplied by the constant 

ratio -., plus the constant f where /is the focal length of the 

objective, and i is the distance between extreme wires. If the 
the distance between the extreme wires be made o.oi of the 
focal length of the objective, then the distance of the stadia- 
rod from the objective (rigidly from F 2 ) is a hundred times the 
intercepted space on the rod, plus the focal length of the ob- 
jective. 

Again, if a base be measured in front of the instrument, 
with its initial point a distance f in front of the object-glass of 
the telescope, then the rod may be held at any point on this 
base-line, and its distance from the initial point, and the space 
intercepted by the extreme wires, will be in constant ratio. 

The lines A % F' and BJ*' in Fig. I show this relation, for 
they are the lines defining the space on the rod which is inter- 
cepted by the extreme wires as the rod moves back and forth. 
Evidently the rod cannot approach so near as F ', for then the 
image would be at an infinite distance behind the lens. Usu- 
ally the extreme position of reticule does not correspond to 
a position of rod nearer than ten to fifteen feet. 

It must be remembered that any motion of the eye-piece, 
with reference to the image and wires, is only made to accom- 



228 SURVEYING. 



modate different eyes, and has no effect in changing the rela- 
tion of wire interval and image. The eye-piece is simply a 
magnifier with which to view the image and wires, but in all 
erecting-instruments it also reinverts the image so as to make 
it appear upright. The effect of the eye-piece has no place in 
the discussion of stadia formulae. 

If the distance of the stadia is to be reckoned from the 
centre of the instrument, which it usually is, and if this dis- 
tance = d, and the distance from the centre of the instrument 
to the objective (CB 2 in Fig. i) = c, then we have, from (3), 

d = A + c = {s+f-\-c. ..... (4) 

Since/", i, and c are constant for any instrument, we may 
measure f and c directly, and then find the value of i by a 
single observation. Proceed as follows : 

1st. Measure the distance from the centre of the instru- 
ment (intersection of plumb-line with telescope) to the objec- 
tive, and call this c. 

2d. Focus the instrument on a distant point, preferably the 
moon or a star, and measure the distance from the plane of 
the cross-wire to the objective, and call this/". 

3d. Set up the instrument, and measure the distance f-\- c 
forward from the plumb-line, and set a mark. From this mark 
as an initial point, measure off any convenient base, as 400 feet. 

4th. Hold the rod at the end of this base, and measure the 
space intercepted by the extreme wires. If we call the length 
of this base b, and the distance intercepted s, then we have, 
from equation (3), 

b=ts, 
1 

or *=jf- (5) 



TOPOGRAPHICAL SURVEYING. 229 

Here we have the value of i in terms of known quantities. 
If it is desirable to set the wires at such a distance apart 

that -7 will be a given ratio, as -3-^-, then i must equal 0.0 if. It 

is possible to set the wires by this means to any scale, so that 
a rod of given length may read any desired maximum distance. 

If it is desired that — should be determined with great ac- 

curacy for a given instrument, with wires already set, so as to 
have a coefficient of reduction for distance, for readings on a 
rod graduated to feet and tenths, for instance, proceed as fol- 
lows : 

Make two sets of observations for distance and intercepted 
interval. The distances should differ widely, as 50 feet and 
500 feet, or 100 feet and 1000 feet, according to the length of 
rod used. The shorter distance should not be less than 50 feet, 
and the longer one not more than 1000 feet with the most 
favorable conditions of the atmosphere. The distances are to 
be measured from the centre of the instrument. Make several 
careful determinations of the wire interval at each position of 
the rod, and take the mean of all the results at each distance, 
and call that the wire interval, s, for that distance, d. We then 
have two equations and two unknown quantities, these latter 

/ 
being — and {f-\- c) in the formula, equation (4), 

<*={s + (f+c). 



f 
Here the d and s are observed, and — and (f-\- c) are found. 

Knowing these, a table could be prepared giving values of d 
for any tabular value of s for that instrument. 

This applies to the reading of distances from levelling-rods. 



230 SURVEYING. 



Some engineers prefer, in this case, to observe the wire 
interval for various measured distances, from the shortest to 
the longest, to be read in practice, and prepare a table by inter- 
polation. If the observed positions are sufficiently numerous, 
this method should give identical results with those obtained 
by the use of the formula. The two methods may be used to 
check each other. 

From equation (4) we see that the distance of the rod from 

the centre of the instrument is a constant ratio (— J times the 

intercepted space on the rod, plus a constant (f-\-c). 

If diagrams or designs be drawn on the stadia-rod to the 

i i 

scale -2, or so that 10 X 7 yards on the rod would correspond 

to 10 yards in distance, and if the rod were decorated with 
symbols of this size, then the distance of the rod from the 
instrument could be read at once by noting how many symbols 
were intercepted between the wires. To this distance must 
then be added the small distance (f-\- c), which is from 10 to 
16 inches' in ordinary field-transits. On all side-readings, taken 
only to locate points on a map, this correction need not be 
added, as one foot is far within the possibilites of plotting. 

201. On the Government Surveys the base is usually 
measured from the centre of the instrument, and its length is 
taken as about a mean of those which the stadia is intended to 
measure, and the symbols scaled by this reading. Then, of 
course, the distance read is always in error by a small amount, 
except when it is the same as the base for which it was gradu- 
ated. For all shorter distances the reading is too small, and 
for all greater distances the reading is too large. Sometimes 
several different lengths of base are taken, as 400, 600, and 
800 feet, all from centre of instrument, and a mean value of 
wire interval used for giving the scale for the diagrams. This 
is practically the same as the other, for in either case the scale 
is correct for but a single distance. 



TOPOGRAPHICAL SURVEYING. 23 1 

The correction to any reading on a stadia so graduated, in 
order to give the distance from the centre of the instrument, is 



*=('+/) (i-#) (6) 



where K = correction, in feet ; 

B = distance read on stadia, in feet ; 
B' = length of base, in feet, for which the stadia was 
graduated. 
If B' — 1000 feet, B = 100 feet, and c +/= 1.5 feet, then 

K- i.5 (-lWir) = + i-35 ^et 

If B had been 2000 feet, then 

K= 1.5 (1 — «#*) = -«-S ^et. 

These corrections are not usually applied. 

202. Another Method of determining the scale for gradu- 
ating the rod is to measure the base from the plumb-line, as 
above, and then, from a fixed point on the lower pait of the 
rod, find the intervals that correspond to various distances, as 
100 feet, 200 feet, 300 feet, etc., and mark these on the board, 
always keeping the lower wire on the fixed, initial point of the 
rod. Then each 100-foot space is subdivided into ten equal 
parts, or symbols ; so that, in reading the rod afterwards, if the 
lower wire is always set on the initial point, the reading always 
gives the correct distance from the centre of the instrument. 

The objection to this method is that the initial point on 
the rod cannot always be seen, on account of obstructions. 

203. Adaptation of Formulae to Inclined Sights. — The 
previous discussion is applicable to horizontal sights only. 



232 



SURVEYING. 



If the rod be held on the top of a hill, and the telescope 
pointed towards it, the reading on the rod will give the linear 
distance from instrument to rod, provided the rod be held per- 
pendicular to the line of sight. As it would be inconvenient to 
do this, let the rod be held vertical in all cases. When the 
line of sight is inclined to the rod, the space intercepted is 




Fig. 62. 



increased in the ratio of 1 to the cos of the angle with the 
horizon. 

Thus, the space A' B' (Fig. 62) for the rod perpendicular 
to the line of sight becomes AB for the rod vertical. But 
A'B' = AB cos v.* 

Let A' B' = r', the reading on the stadia for perpendicular 
position ; and 

Let AB = r, the actual reading obtained for a vertical 
position. 

Then r' = r cos v. 

f 

But in equation (4) we have — s= r\ and therefore r f -f- c 

It 



*This assumes that A'B' is perpendicular to CB and CA, which it is practi- 
cally, since the angle A CO' is so very small, usually about 15'. 



TOPOGRAPHICAL SURVEYING. 233 

-j~yis the distance C0'\ whereas the distance on the horizon- 
tal, CO, is generally desired, and for this we have 

CO = d= CO' cos v — (r' + c+f) cos v 

= r cos 3 v -f- (c +/) cos v, (7) 

This is the equation for reducing all readings on the stadia 
to the corresponding horizontal distances. 

The vertical distance of O' above O is equal to CO' sin V. 



But CO' = r' +/+ c — r cos v +/+ c, 

hence 

00' = h = r cos v sin v -f- {f-\- c) sin v 

= \r sin 2v + (/+ c) sin v. (8) 

Equation (8) is used for finding the elevation of the point 
on which the stadia is held above or below the instrument sta- 
tion. 

204. Table V. gives the values d and h computed from 
these formulae for a stadia reading of 100 feet (or metres, or 
yards), with varying angles up to 30 . 

It will be noted that the second term in the right member 
of equations (7) and (8) is always small, and its value depends 
on the instrument used. The values of this term are taken 
out separately in the table ; and three sets of values are given 
of {c-\-f), — viz., 0.75 feet, 1.00 feet, and 1.25 feet. If the 
work does not require great accuracy, these small corrections 
may be omitted. 

The use of the table directly involves a multiplication for 



234 



SURVEYING. 



every result obtained. Thus, if the stadia reads 460 feet, the 
angle of inclination 6° 20', and we have/-|-£ = 1 foot, then 



and 



d = 4.60 X 98.78 + 0.99 = 455.4 feet, 
h — 4.60 X 10.96 -J- 0.1 1 = 50.53 feet. 



The table is not generally used for reductions for d when the 
angle of elevation is less than 3 to 5 degrees. When v = 5 
44', this reduction amounts to just one per cent. When an 
error of 1 in 100 can be allowed, then the reduction to the 
horizontal would not be used under 6°. If the second term in 
c-\- fbe. also neglected, these two errors tend to compensate ; 
and if c -\- f iox the instrument used is I foot, and both these 
corrections be omitted, they do exactly compensate when the 



stadia read 


ing 


is 


100 feet, vertical ar 


igle 


5° 44'. 


<< a 






200 " " 




4° 04'. 


<< «< 






300 " " 




3° 20'. 


a a 






400 " " 




2° 52'. 


a u 






500 




2° 32'. 


a a 






1000 " " 




i° 4 6 / . 


it a 






2000 " " 




i° 18'. 



Therefore the reduction to the horizontal need never be 
made when v is less than 2°, and it generally may be neglected 
when v is less than 6°. 

In obtaining the difference of elevation, h, the term in 
c -\- f may be omitted for all angles under 6° if errors of o.I 
foot are not important. For elevations on the main line, how- 
ever, this term should always be included. 

In practice, therefore, the tables are mostly used to obtain 
the difference of elevation from the given stadia reading and 
angle of elevation. 



TOPOGRAPHICAL SURVEYING. 235 

205. Reduction Diagram. — Since the use of these tables 
involves a multiplication each time, and since a table for vary- 
ing distances and angles would be very voluminous, it is prefer- 
able to take out the elevations from a diagram. Such a diagram 
has been prepared, to be used in place of the table. It is ar- 
ranged with both coordinates in feet, but can be used for both 
coordinates in metres, since the same unit is used for both. 
It will only be neccessary to re-number the divisions, to adapt 
it to the new scale. 

This diagram has been prepared with great care, and is 
arranged to give distances to 500 yards or metres, or 1 500 feet, 
with elevations to 50 feet. For longer distances or higher 
elevations for a single pointing, the results may be obtained 
from the table. Elevations are taken off from the diagram 
to the nearest tenth of a foot, with great readiness ; as the 
smallest spaces are 2 millimetres square, and these correspond 
to two-tenths of a foot in elevation. It is of more convenient 
use than extended tables, and is just as accurate ; the nearest 
tenth of a foot being quite as exact as one is warranted in 
writing elevations when obtained in this manner. 

Corrections to the distances read are also obtained from 
this diagram for large vertical angles.* 

THE INSTRUMENTS. 

206. The Transit. — That the transit may be best adapted 
to this work, there are certain features it should possess, 
though all of them are by no means essential. They will be 
named in the order of their importance. 

1st. The horizontal limb should be graduated from zero to 
360 , preferably in the direction of the movement of the hands 
of a watch. 

* The diagram is printed on heavy lithographic paper 20 by 24 inches, from 
an engraved plate, and can be had from the publishers of this volume. Price 
50 cents, post paid. 



236 SURVEYING. 



2d. The instrument should have a vertical circle rigidly at- 
tached to the telescope axis, and not simply an arm that is 
fastened by a clamp-screw, and which reads on a fixed arc be- 
low. So much depends on the vertical circle holding its adjust 
ment that its arrangement should be the best possible. Since 
the telescope is not transited, the vertical circle need not be 
complete. 

3d. The telescope should be inverting, for two reasons : 
first, in order to dispense with two of the lenses, and so obtain 
a better definition of image ; and, second, that the objective 
may have a longer focal length, thus giving a flatter image and 
a less distorted field. 

4th. The stadia wires should be fixed instead of adjustable, 
as in the latter case they are not stable enough to be reliable. 

5th. The bubbles on the plate of the instrument should be 
rather delicate, so that a slight change in level may become 
apparent. They should also hold their adjustments well. This 
is very important, in order that the readings of the vertical 
angles may be reliable. It is also of great importance in 
carrying azimuth where the stations are not on the same level. 

6th. The horizontal circle should read to thirty seconds ; 
and there should be no eccentricity, so that one vernier-read- 
ing shall be practically as good as two. 

7th. The instrument (or tripod) should have an adjustable 
centre, for convenience of setting over points. 

8th. A solar attachment to the telescope will be found very 
convenient. In most regions the azimuth can be checked up 
by the reading of the needle, but in many places this is not 
reliable. 

207. Setting the Cross-wires. — The engineer should al- 
ways have at hand a spider's cocoon of good wires, and a small 
bottle of thick shellac varnish. If the dry shellac is carried it 
may be dissolved in alcohol. If no such cocoon is at hand a 
spider may be caught and made to spin a web. The small, 



TOPOGRAPHICAL SURVEYING. 237 

black, out-door spider makes a good web for stadia purposes. 
A new wire should be allowed to dry for a few minutes, and an 
old one should be steamed to make it more elastic. The 
wires for stadia-work should be small, round, and opaque. 
Some wires are translucent, and some are flat and twisted like 
an auger-shank. 

Scratches must be made across the face of the reticule 
where the wires are to lie. These must be made with great 
care, so as to have them equally spaced from the middle wire, 
parallel to each other, and perpendicular to the vertical wire. 
The distance apart of the extreme wires is to be computed by 
equation (5) for any desired scale on the rod. > 

Take a piece of web on the points of a pair of dividers, by 
wrapping the ends several times about the points, which should 
be separated by about an inch ; stretch the wire, by spreading 
the dividers, as much as it will bear ; and lay the dividers 
across the reticule in such a way that the web comes in place. 
The dividers must be supported underneath, so that the points 
will drop just a trifle below the top of the reticule ; otherwise 
they would break the web. Move the dividers until the web 
is seen, by the aid of a magnifying-glass (the eye-piece will do), 
to be in exact position. Then take a little shellac on the end 
of a small stick or brush, and touch the reticule over the web, 
being careful to have no lateral motion in the movement. 
The shellac will harden in a few minutes, when the dividers 
may be removed. Shellac is not soluble in water. 

208. Graduating the Stadia-rod. — The stadia-rod is 
usually a board one inch thick, four or five inches wide, and 
twelve to fourteen feet long. Sometimes this is stiffened by a 
piece on the back. To graduate the rod, it is necessary to 
know what space on the rod corresponds to a hundred feet (or 
yards, or metres) in distance. Either of the three methods 
cited on pp. 7-8 may be used for doing this, but the first is rec- 
ommended. Thus, measure off c +/in front of the plumb- 



238 SURVEYING. 



line, and set a point. From this point measure off any con- 
venient base, as 200 yards, on level ground, and hold the blank 
rod (which has had at least two coats of white paint), at the end 
of this base-line. Have a fixed mark or target on the upper 
part of the rod, on which the upper wire is set. Have an assist- 
ant record the position of the lower wire as he is directed by 
the observer. Some sort of an open target is good for this pur- 
pose, but any scheme is sufficient that will enable the observer 
to fix the position of the extreme wires at the same moment with 
exactness. This work should be done when there is no wind, 
and when the atmosphere is very steady : a calm, cloudy day 
is best. Repeat the operation until the number of results, or 
their accordance, shows that the mean will give a good result. 
If the base was 200 yards long, divide this space into two equal 
parts, then each of these parts into ten smaller parts, and 
finally each small space into five equal parts; and one of 
these last divisions represents two yards in distance. Dia- 
grams are then to be constructed on this scale, in such a way 
that the number of symbols can be readily estimated at the 
greatest distance at which the rod is to be read. The individ- 
ual symbols should be at least three inches across ; so that, if 
one of these is to represent ten units, as yards or metres, then 
IOO units will cover 2\ feet, and a rod 14 feet long will read a 
distance of 560 units (yards or metres). If it is desired to read 
distances of a quarter of a mile or more, the rod should be 
graduated to read to yards (or five-foot units, or metres) ; but 
if it is not to be used for distances over 500 to 1000 feet, it 
might be graduated to read to feet. This question must be 
decided before the wires are set, and then they must be spaced 
accordingly. 

In measuring the base, care should be taken to test the 
chain or tape carefully by some standard. 

If the rod is to be graduated to read to feet, of course 
the base should be some even hundreds of feet, as 600. 



TOPOGRAPHICAL SUR VE YING. 



2 39 



In Fig. 63 are shown four designs for stadia-rods which 
have been long in use, and are found to work well. They are 
intended to be all in black on a white ground.* It will be 
noticed that the shortest lines in these diagrams all cover a 
space of two units on the rod. In diagrams 2 and 3 the units 
are either yards or metres, while in I they are units of five 
feet each. In diagram 4 the units are of two feet each. The 



Zoo 



aoo 



too 




Fig. 63. 



successive units are found at the middles and limits of these 
lines and spaces. Wherever the wire falls, there should be a 
white ground on some part of the cross-section ; and the more 
white ground the better, provided the figures are distinct. 
The black paint may be put on heavy, so that one coat will be 
sufficient. 

The 50- and 100-unit marks should be distinguished by 
special designs. There should usually be at least two boards 
with each instrument, and sometimes three and four are needed. 
Of course, these are all duplicates. After the unit scale is 
obtained, or the space on the rod corresponding to a hundred 

* Some engineers prefer red on the 100-unit figures. 



240 SURVEYING. 



units in distance, these ioo-unit spaces should be so distributed 
as to be symmetrical with reference to the ends of the rod. The 
reason of this will appear later. Having determined how many 
ioo-unit spaces there will be on the rod, fix the position of the 
two end ioo-unit symbols with reference to this symmetry, and 
then the rod is subdivided from these points. 

Special pains should be taken to have the angular points of 
the diagrams well defined and in position. These points are 
on the lines of subdivision of the rod. 

After one rod is subdivided, the others of that set may be 
laid alongside, and all fastened rigidly together ; and then, by 
means of a try-square or T-square, the remaining rods may be 
marked off. 

The wire interval should be tested every few months by 
remeasuring a base, as was done for graduation, and reading 
the rod on it, to see if this shows the true measured distance. 
This is to provide against a possible change in the value of the 
wire interval. If the wires are stretched reasonably tight when 
they are put in, they seldom change, If they are too loose, 
they swell in wet weather, and may sag some. The reticule 
should be so firm that the variable strain on the adjusting- 
screws will not distort it appreciably. 

If the wire interval is found to have changed, either the 
rods must be regraduated, or else a correction must be made 
to all readings of importance. What are called the " side 
shots," which make up a large proportion of the readings 
taken, would not need to be corrected. 

If the wires are adjustable, any unit scale may be chosen 
at pleasure, and the wires adjusted to this scale. Then, if the 
intervals change, the matter is corrected by adjusting the 
wires. The adjustable wires are generally used to obtain dis- 
tances from levelling-rods, where it is desirable that each foot 
on the rod shall correspond to a hundred feet in distance. For 
the ordinary stadia-rods, fixed wires are preferable. 



TOPOGRAPHICAL SURVEYING. 241 



GENERAL TOPOGRAPHICAL SURVEYING. 

209. The Topography of a region includes not only the 
character and geographical distribution of the surface-cover- 
ing, but also the exact configuration of that surface with 
reference to its elevations and depressions. Thus any point 
is geographically located when its position with reference to 
any chosen point and a meridian through it is found, but to 
be topographically located its elevation above a chosen level 
surface must also be known. A topographical survey consists 
in locating by means of three coordinates a sufficient number 
of points to enable the intervening surface to be known or 
inferred from these. Evidently the points chosen should be 
such as would give the greatest amount of information. As 
for geographical outline, the corners, turns, or other critical 
points are chosen, so for configuration the points of changes 
in slope, as the tops of ridges and bottoms of ravines, or the 
brow and foot of a hill, are chosen as giving the greatest 
information. 

210. Field-work. — Let it be required to make a topo- 
graphical survey of either a small tract, a continuous shore- 
line, or of a large area, for the purpose of making a contour 
map. 

In case of the small tract, any point may be taken as a 
point of reference, and the survey referred to it as an origin. 
In case of an extended region, a series of points should be 
determined with reference to each other, both in geographical 
position and in elevation. These determined points should 
not be more than about three miles apart. The points of ele- 
vation or bench-marks need not be identical with those fixed 
in geographical position. These last are best determined by a 
system of triangulation, and are called" triangulation stations." 
In the succeeding discussion, the symbol A will be used for 
triangulation station, and B.M. for bench-mark, 
16 



242 • SURVEYING. 



First, a system of triangulation points is established, the 
angles observed, azimuths and distances computed, and the 
stations plotted to scale on the sheet which is to contain the 
map. This plotting is best done, for small areas, by comput- 
ing the rectangular coordinates (latitudes and departures), 
and plotting them from fixed lines which have been drawn 
upon the map, accurately dividing it into squares of iooo or 
5000 units on a side. They may, however, be plotted directly 
from the polar coordinates (azimuth and distance) as given by 
the triangulation reduction. For this purpose, the sheet on 
which the map is first drawn, called the field sheet, should have 
a protractor circle printed upon it, about twelve inches in diam- 
eter. These protractor sheets of drawing-paper can be obtained 
of most dealers in drawing-materials, or the protractor circle 
may be printed to order on any given size or quality of paper.* 
These protractor circles are very accurate,, and are graduated 
to 15' of arc. Plotting can be done to about the nearest 5'. 

Second, a line of levels is run, leaving B.M.'s at convenient 
points whose elevation are computed, all referred to a com- 
mon datum. If the A's are not also B.M.'s, then a B.M. 
should be left in the near vicinity of each A. This is not 
essential, however. 

Third, the topographical survey is then made, and referred 
to, or hung upon, this skeleton system of A's and B.M.'s. 

The topographical party should consist of the observer, a 
recorder, two or three stadia-men, and as many axemen as 
may be necessary, generally not more than two. 

The azimuth, preferably referred to the true meridian, is 
known for every line joining two A's, as well as the length of 
such line. 

Set up the transit over a A, and set the horizontal circle 



* Messrs. Queen & Co.. Philadelphia, or Blattner & Adam of St. Louis, can 
furnish such sheets. 



TOPOGRAPHICAL SURVEYING. 243 

(which should be graduated continuously from o° to 360 in 
the direction of the hands of a watch) so that vernier A will 
read the same as the azimuth of the triangulation line by which 
the instrument is to be oriented. Clamp the plates in this 
position, and set the telescope to read on the distant A. Now 
clamp the instrument below, so as to fix the horizontal limb, 
and unclamp above. The azimuths of the triangulation lines 
are generally referred to the south point as the zero, and in 
small systems of this sort the forward and back azimuths are 
taken to be 180 apart. When the instrument has been set 
and clamped, all subsequent readings taken at that station are 
given in azimuth by the readings of vernier A on the horizon- 
tal limb. For any pointing, therefore, the reading of this 
vernier gives the azimuth of the point referred to the true 
meridian, and the rod reading gives the distance of the point 
from the instrument station. These enable the point to be 
plotted on the map. To draw the contour lines, elevations 
must also be known. 

If the elevation of the A is known, measure the height of 
instrument (centre of telescope) above the A on the stadia,* as 
soon as the instrument is levelled up over that station. Sup- 
pose this comes to the 212-unit mark. Write in the note-book, 
as a part of the general heading for that station, " Ht. of Inst. 
= 212." Then, for all readings from that station for eleva- 
tions, bring the middle horizontal wire to the 212-unit mark 
on the rod, and read the vertical angle. From this inclination 
and distance, the height of the point above or below the 
instrument station is found. If the rod be graduated sym- 
metrically with reference to the two ends, one need not be 
careful always to keep the same end down, and so errors from 
this cause are avoided. 

* Or, if preferred, a light staff, about five feet long, may be carried with the 
instrument for this purpose, it being graduated the same as the stadia rods for 
this instrument. 



244 SURVEYING. 



The record in the note-book consists of — 

1st. A Description of the Point, as, " N.E. cor. of house," 
" intersec. of roads," "top of bank," " C.P." for "contour 
point," which is taken only to assist in drawing the contours, 
" 16 " for " stadia station 16," etc. 

2d. Reading of Ver. A. 

3d. Distance. 

4th. Vert. Angle. 

These four columns are all that are used in the field. 
There should be two additional columns on the left-hand page, 
for reductions, viz. : 

5th. Difference of elevation, corresponding to the given 
vertical angle and distance, and which is taken from a table or 
diagram. 

6th. Elevation, which is the true elevation of each point 
referred to the common datum. 

The right-hand page should be reserved for sketching. 

It will be found most convenient to let the sketching pro- 
ceed from the bottom to the top of the page ; as in this case 
the recorder can have his book properly oriented as he holds 
it open before him, and looks forward along the line. The 
notes may advance from top to bottom, or vice versa, as de- 
sired. If there are many " side shots" from each instrument 
station, one page will not usually contain the notes for more 
than two stations, and sometimes not even for one. 

The sketch is simply to aid the engineer when he comes to 
plot the work, and may often be omitted altogether. One 
soon becomes accustomed to impressing the characteristics of 
a landscape on his memory so as to be able to interpret his 
notes almost as well as though he had made elaborate sketches. 
For beginners the sketches should be made with care. The 
observer should usually make his own sketches and plot his 
own work. 

After the instrument is oriented over a station, and its 



TOPOGRAPHICAL SURVEYING. 245 

height taken on the stadia, the stadia-men go about holding 
the rods at all points which are to be plotted on the map, 
either in position or in elevation, or both. The choice of points 
depends altogether on the character of the survey ; but since 
a single holding of the rod gives the three coordinates of any 
point within a radius of a quarter of a mile, it is evident the 
method is complete, and that all necessary information can 
thus be obtained. For very long sights, the partial wire inter- 
vals (intervals between an extreme and the middle wire) may 
be read separately on the stadia, and in this way twice as great 
a distance read as the rod was designed for. The limit of 
good reading is, however, usually determined by the state of 
the atmosphere, rather than by the length of the rod. When 
the air is very tremulous, good readings cannot be made over 
distances greater than 500 feet ; while, when the atmosphere 
is very steady, a half-mile may be read with equal facility. 

Before the instrument is removed from the first station, 
the forward stadia-man selects a suitable site for the next 
instrument station (generally called stadia station, and marked 
H, to distinguish it from a triangulation station, A), and drives 
a peg or hub at this point. This peg is to be marked in red 
chalk, with its proper number, and should have a taller mark- 
ing-stake driven by the side of it. The peg for the El should 
be large enough to be stable ; for it must serve as a reference 
point, both in position and elevation, during the period of the 
survey. It is often desirable to start a branch line, or to 
duplicate some portion of the work, with one of these stations 
as the starting-point ; and, since each H is determined, in 
position and elevation, with reference to all the others, one 
can start a branch line from one of these as readily as from a 
A. It is not usually necessary to put a tack in the top, but 
the centre may be taken as the point of reference. The stadia- 
man first holds his stadia carefully over the centre of this H, 
with its edge towards the instrument, so as to enable the 



246 SURVEYING. 



observer to get a more accurate setting for azimuth. The 
observer could just as well bisect the face of the rod ; but, if 
held in this position, the centre of the rod may not be so 
nearly over the centre of the peg as when held edgewise. 
This holding of the rod edgewise for azimuth checks the care- 
lessness of the stadia-man, and is done only for readings on 
instrument stations. 

At a signal from the observer, the stadia is turned with its 
face to the instrument, and the observer reads the distance and 
vertical angle. 

It is advisable, in good work, to re-orient and relevel the 
instrument just before reading to the forward EL The transit 
is very apt to get out of level after being used for some time, 
with more or less stepping around it, and the limb may have 
shifted slightly on the axis, both of which might be so slight 
as to make no material difference for the side readings, but 
which would be important in the continued line itself. It is 
best, therefore, to level up again, and reset on the back station, 
before reading to the forward one. If it is inconvenient for 
the rear rodman to go back to this station to give a reading, a 
visible mark should be left there, to enable the observer to 
reset upon it for azimuth, as it is not necessary to read distance 
and vertical angle again. 

When the instrument is moved, it is set up over the new 
station, and the new height of instrument determined and 
recorded. The rear stadia-man is now holding his rod, edge- 
wise, on the station just left ; and by this the observer orients 
his instrument, making vernier A read 180 different front its 
previous reading on this line. Clamping the plates at this 
reading, the telescope is turned upon the rod on the back sta- 
tion, and the lower plate clamped for this position. The circle 
is now oriented, so that, for a zero-reading of vernier A, the 
telescope points south. 

It will be noted that the telescope is never reversed in this work. 



TOPOGRAPHICAL SURVEYING. 247 

The distance and vertical angle should both be reread, on 
this back reading, for a check. If the vertical circle is not in 
exact adjustment, this second reading of the vertical angle will 
show it, for the numerical value of the angle should be the 
same, with the opposite sign. If they are not the same, then 
the numerical mean of the two is the true angle of elevation, 
and the difference between this and the real readings is the 
index error of the vertical circle. This error may be corrected 
in the reduction, or the vernier on the vertical circle may be 
adjusted. 

The second reading of the vertical angle on the stadia- 
stakes is thus seen to furnish a constant check on the adjust- 
ment of the vertical circle, and should therefore never be 
neglected. If the circle is out of adjustment by a small 
amount, as one minute or less, in ordinary work it would not 
be necessary either to adjust it or to correct the readings on 
side-shots, for the elevations of contour points are not required 
with such extreme accuracy. The mean of the two readings 
on stadia-stakes would still give the true difference of elevation 
between them, so that there would be no continued error in 
the work. * 

The work proceeds in this manner until the next A is 
reached. In coming to this station, it is treated exactly as 
though it were a new El; and the forward reading to it, and 
the back reading from it, are identical with those of any two 
consecutive El's. Having thus occupied the second A, and 
having oriented the instrument by the last El, turn the tele- 
scope upon some other A whose azimuth from this one is 
known. The reading of vernier A for this pointing should be 
this azimuth, and the difference between this reading and the 
known azimuth of the line is the accumulated error in azimuth 
due to carrying it over the stadia line. This error should not 
exceed five minutes in the course of two or three miles in good 
work. 



248 SUR VE YING. 



The check in distance is to be found from plotting the line, 
or from computing the coordinates of the single triangulation 
line, and also of the meandered line, and comparing the re- 
sults. 

The elevations are checked by computing the elevation of 
the new A from the stadia line, and comparing this with the 
known elevation from the line of levels. 

In case the elevations of the A's are not given, but only- 
certain B.M.'s in their vicinity, then the check can be made 
on these just the same. Thus, in starting, read the stadia on 
the neighboring B.M., and from this vertical angle compute the 
elevation of the A over which the instrument sets, and then 
proceed as before. In a similar manner, the check for eleva- 
tion at the end of the line may be made on a B.M. as well as 
on the A. 

A quick observer will keep two or three stadia-men busy 
giving him points ; so that in flat, open country, with long 
sights, it may be advisable to have three or even four stadia- 
men for each instrument. In hilly country more time will be 
required in making the sketches, and hence fewer stadia-men 
are required. 

After the instrument is oriented at each new station, the 
needle should be read as a check. To make this needle-read- 
ing agree with the readings of the verniers on the horizontal 
circle (the north end with vernier A, and the south end with 
vernier B, for instance), graduate an annular paper disk the 
size of the needle-circle, and figure it continuously from o° to 
360 , in the reverse direction to that on the horizontal limb of 
the instrument, and paste it on the graduated needle-circle in 
such a position that the north end of the needle reads zero 
when the telescope is pointing south. If the variation is 6° 
east, this will bring the zero of the paper scale 6° east of south 
on the needle-circle. This position of the paper circle is then 
good within the region of this variation of the needle. When 



TOPOGRAPHICAL SURVEYING. 249 

the survey extends into a region where the variation is differ- 
ent, the scale will have to be reset. 

With these conditions, when the instrument is oriented for 
a zero-reading when the telescope is south, the reading of the 
north end of the needle will always agree with the reading of 
vernier A, and the south end with vernier B. It is so easy a 
matter to let the needle down, and examine at each H to see if 
this be so, that it well pays the trouble. No record need be 
made of this reading, as it is only used to check large errors. 

211. Reducing the Notes. — The only /eduction necessary 
on the notes is to find the elevation of all the points taken, with 
reference to the fixed datum, and sometimes to correct the 
distance read on th% rod for inclined sights. The difference of 
elevation between the H and any point read to, as well as 
the correction to the horizontal distance, can be taken from 
Table V. or from the diagram. The methods of using these 
have been explained (see pp. 234-5). After the differences of 
elevation are taken out, the final elevations of the points are 
to be computed by adding algebraically the difference of eleva- 
tion to the elevation of EL 

The following is a sample page with these reductions: 



250 



SURVEYING. 



April 20, 1883. 
At 4. Ht. of Inst. = 87. 



Gazzam, Observer. 
Baier, Recorder. 

Elevation == 24'. 94. 



Object. 



03 

Bridge 

S. E. cor. of house 

On road 

Water-level, foot of hill. 

05 

C.P 



Azimuth. 
Ver. A. 


Distance. 


Vert. 
Angle. 


Difference 

of 
Elevation. 




yds. 






328° 10' 


199 


— 0° io' 


- i'-56 


127° 40' 


70 


4-o° 32' 


+ i'.9 


142° 35' 


90 


+ o° 15' 


+ i'.a 


180 25' 


114 


+0° 7' 


+ o'.y 


230 15' 


224 


T°°57' 


— 10'. 9 


128 33' 3o" 


2l6 


+ o°55' 


+10'. 38 


190 48' 


2IO 


-J- I 2 


+n'.4 



Eleva- 
tion 
above 
Datum. 



26'. 8 
26'. I 
25'.6 

14'. o 

36'. 3 



At 5. Ht. of Inst. = 78. Mean = + io'.26. 35'.20. 



04 

S. W. cor. of house 

Edge of bank 

S.E. cor. of R.R. station.. 
Railroad track 



06 



308 33' 30" 


215 


- o° 54' 


— 10'. 13 


43° 3o' 


104 


+ 3° 3' 


+16'. 


332° 10' 


98 


+ i°57' 


-f-10'.i 


85° 3o' 


158 


-f-1 2' 


+ 8'. 5 


43° 55' 


40 


+ 2° 53' 


+ 6'.o 


79° 30' 


270 


+ o° 9' 


4~ 2 '- 1 


79° 3o' 


200 


-o° 2' 


— o'.36 



51.2 

45'-3 
43'- 7 
41'. 2 

37'-3 



At 6. Ht. of Inst. = 79. Mean 



C/.54. 34'.66. 



05 

Cor. of house. 
Top of hill . . . 
Wagon road.. 

08 

C.P 

7 



259 30 

277° 55' 
87° 25' 
58° 15' 

40° 37' 

41° 45' 
5° 25' 



200 


+ o° 4' 


+ o'-72 


112 


+ 3° 26' 


+i9'-7 


I98 


4 4° 48' 


+49'- 3 


186 


+ 4° 25' 


+42'. 9 


216—3 
213 


+ 6° 33' 


+73'-53 


III 


+ 4° 41' 


+27'. 


194 


+ o° 12' 


+ 2'.04 



54.4 

84'.o 
77'.6 



61'. 7 



TOPOGRAPHICAL SURVEYING. 2$l 

It will be noted that the reading on B 5 from B 4 has a 
distance of 216 yards, and a vertical angle of +°° 55' i while 
on the back reading, from B 5 to B 4 the distance is 215 yards, 
and the vertical angle — 0° 54'. The distance was probably 
between 215 and 216 yards, and the vertical circle was prob- 
ably slightly out of adjustment. The difference of elevation 
is taken out for both cases, however, being respectively 10.38 
feet and 10.13 feet. The mean of these is 10.26 feet, which 
stands as a part of the general heading at Q 5. The true 
elevation of 5 is then found by adding 10.26 to 24.94, 
giving 35.20 feet, which is also set down as part of the general 
heading. 

The elevations on the side-readings from this station can 
now be taken out. These side-elevations are only used for 
obtaining the contours, and hence are only taken out to tenths 
of a foot. When the contours are ten feet apart or more, 
these side-elevations need only be taken out to the nearest 
foot. The elevations of the stadia stations should, however, 
always be taken out to hundredths, to prevent an accumula- 
tion of errors in the line. 

The reduction for distance may also be taken from that 
portion of the diagram arranged for this purpose. This is 
used the same as the other portion ; and the correction is 
found, which is to be always subtracted from the rod-reading. 
Thus, in the reading on 8 from B 6, we have a reading of 
216 yards, and a vertical angle of 6° 33'. The correction here 
is 2.16 X 1-3 = 2.8 yards, as found from the table. Calling this 
3 yards it is subtracted from the 216, leaving 213 yards as 
the distance to be plotted. It is only the stadia-line distances 
that need ever be corrected in this way, the corrections being 
usually so small that it is not important on the side-shots. 

It will be noted that two El's were set from H 6. This 
was done because a branch-line was run from E 6 over the 
bluffs. In order to make it unnecessary to occupy E 6 again 



252 SURVEYING. 



when the branch-line came to be run, 8 was set while H 6 
was occupied in the main-line work. When the branch-line 
came to be run, the instrument was taken directly to H 8, and 
oriented on H 6 by the readings previously taken from 6. 

The right-hand page of the note-book, opposite the notes 
given above, is occupied with a sketch of the locality, with the 
El's marked on, the general direction of the contour lines, the 
railroad, stream, houses, etc.* 

212. Plotting the Stadia Line. — It is customary to first 
plot the stadia stations alone, from one El to the next, to find 
whether or not it checks within reasonable limits. This part 
of the work should be done with extreme care, so that if it 
does not check it cannot be attributed to the plotting. In 
case it does not check within the desired limit, then the line of 
investigation will be about as follows until the error is found : 

1st. Replot the stadia line. 

2d. Recompute and replot the triangulation line. 

3d. By examining the discrepancy on the plot, try and 
decide whether the error is in azimuth or distance, and, if 
possible, where such error occurred, and its amount. 

4th. Examine the note-book carefully, and see if there is any 
evidence of error there. 

5th. If there is a large probability that the error is of a 
certain character, and that it occurred at a certain place, take 
the instrument to that station, set it up, and redetermine the 
azimuths or distances which seem to be in error. 

6th. If there is no high probability of any certain errors to 
be examined for in this way, then go back and run the line 
over, taking readings on \H's only. If the elevations had been 
found to check, the vertical angles may be omitted on this 
duplicate line ; and, on the other hand, if the plot came out all 
right, but the elevations could not be made to check, then a 
duplicate line must be run to determine this alone ; and in this 

* These notes were taken from a field-book of a topographical survey of 
Creve Cceur Lake by the engineering students of Washington University. 



TOPOGRAPHICAL SURVEYING. 253 

case the vertical angles between E's are all that need be read. 
In cases of this kind, it will be found a great help to have the 
El's so well marked that they can be readily found. 

With reasonable care in reading and in the handling of the 
instrument, it will never be necessary to duplicate a line entire, 
for all readings between Q's are checked. The vertical angles 
and distances are checked by reading them forward and back 
over every stadia line ; and the azimuth is checked by the 
needle readings, and also when the second A is reached. 

If, in the progress of the work, the readings on the back H 
for distance and vertical angle do not fairly agree with these 
quantities as read from the previous station, the recorder 
should note the fact : and the observer should then re-examine 
these readings; and, if found to be right, the first readings, 
taken from the other station, should be questioned, and the 
mean not taken in the reduction. 

For plotting the stadia lines a parallel ruler (moving on 
rollers) is very desirable ; otherwise, triangles must be used. 
The plotting is done by setting the parallel ruler or triangle 
on the proper azimuth as found from the protractor printed on 
the sheet, moving it parallel to itself to the station from which 
the point is to be plotted, and drawing a pencil line in the right 
direction. Then, with a triangular scale, — or, better, with a 
pair of dividers and a scale of equal parts, — lay off the correct 
distance on this line ; and this gives the point. 

If the instrument was oriented in the field for a zero read- 
ing for a south pointing, then the protractor on the sheet must 
have its south point marked zero, and increase around to 360 
in the same direction in which the limb of the instrument in- 
creases, preferably in the direction of the movement of the 
hands of a watch. 

213. Check Readings. — To enable the observer to locate 
large errors in azimuth or distance, or both, it is a good prac- 
tice to take azimuth readings to a common object from a series 
of consecutive stations, if such be possible. If the plot does 



254 SURVEYING. 



not close, go back and plot in these azimuths; and if there has 
been no error in azimuth or distance between El's, and no error 
in reading the azimuths for these pointings, then all these lines 
will meet in a common point on the plot. If all but one in- 
termediate line meet at a point, then the error probably was 
in reading the azimuth of this pointing alone. If several of 
the first pointings intersect in a point, and the remaining point- 
ings of the set taken to this object intersect in another point, 
then it is highly probable that the error was in reading the 
azimuth or distance of the line connecting these two sets of 
El's ; and the relative position of the points of intersection 
will enable the observer to decide whether the error was in 
azimuth or distance, and about how much. If, in this way, 
the error be located, the instrument can be taken to this point, 
and the readings retaken. 

214. Plotting the Side-readings. — Having plotted the 
stadia line and made it check, the next step is to go back and 
plot in the side-readings. For doing this, a much more rapid 
method may be used than that described above. 

Divide the sheet into squares by horizontal and vertical 
lines spaced uniformly at from 1000 to 5000 units apart, ac- 
cording to scale. These lines are to be used for orienting the 
auxiliary protractor, and also to test the paper for stretch or 
shrinkage. 

The side-readings are now plotted by the aid of a paper 
protractor, such as is shown in Fig. 64, This is made from a 
regular field-protractor sheet. The graduated circle printed 
on the sheet is used ; and this is some 12 inches in diameter, 
and graduated to 15 minutes. The sheet is trimmed down to 
near the graduated circle, and the edges divided, as shown in 
the figure, to any convenient small scale.* This sheet is to be 

* It is sometimes desirable to make the open space DFE rectangular and 
graduate the sides of the space ABF instead of the outer edges. The pro- 
tractor can then be used nearer the edge of the sheet. 



TOPOGRAPHICAL SUP VE YING. 



255 



laid upon the plot, with its centre, C, coinciding with the 0. 
It is oriented by bringing the corresponding spaces on opposite 
edges to coincide with any one of the spaced lines on the 
plot. This circle then has its position parallel to that of the 
protractor circle printed on the sheet, and an azimuth taken 
from the one will agree with an azimuth taken from the 
other. When this auxiliary protractor has been so centred 
and oriented, let it be held in place by weights. Now the 
part ADEB folds back, on the line AB, into the position indi- 
cated by the dotted lines. The portion DEF is cut out en- 



1 1 1 1 1 1 1 1 1 1 1 1 1 1 | 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 mraxrr 




MM I "M I MM I llll I II II I II I ll M M I I Ml I M M I II II I I I I I l l l ll 



Fig. 64. 



tirely, so that when the flap is turned back the space AFB 
is left open. This space is to be large enough to include the 
longest side-readings when plotted to scale ; that is, the radius, 
CF, of the circle to the scale of the drawing must exceed the 
longest readings. We now have a protractor circle about the 
H, with this station for its centre. 

Take a triangular scale, select the side to be used in laying 
off the distances, and paste a piece of strong paper on the 
lower side at the zero point. Make a needle-hole through this 



2 $6 SURVEYING. 



paper close to the edge, at the zero of the scale. Fasten a 
needle through this hole into the point which marks the exact 
position of the EJ. The scale can now swing freely around the 
needle, on the auxiliary protractor ; and its zero remains at 
the centre of the station from which the points are to be 
plotted. 

To plot any point, swing the scale around to the proper 
azimuth, and at the proper distance mark with the pencil the 
position of the point. If this marks a feature of the land- 
scape, it should be drawn in at once, before going farther ; and 
if the elevation of the point will be needed in sketching the 
contours, this should also be written in. For contour points, 
the elevation is all that is put down. 

In this manner the points can be plotted very rapidly* A 
six-inch triangular scale, divided decimally, will be found best 
for this. 

If there is very much of this work to be done, it might be 
found advisable to have a special scale constructed for the 



d 



|i | iHH i ii 1 1 l li | i il i| i i i i |iiii |i m i i ii i |ii i i|i i ii| i in i r i n 



Fig. 65. 



purpose. Fig. 65 is one form of such a scale drawn one-third 
size, which would be found very convenient and cheap. It 
should be graduated on a bevel edge, and to such a scale that 
the units of distance used on the rod may be plotted to the 
scale of the drawing. The small needle-hole, in line with the 
graduated edge, should be only large enough to fit the 
needle-point used, so that there would be no* play. The rule 
then turns on an accurate centre, which will not wear. Such 
scales, six inches long, could be constructed very cheaply of 
German silver by any instrument-maker. 



TOPOGRAPHICAL SUR VE YING. 



257 



A special form of protractor shown in Fig. 66* has also 

been used with great success on the Mississippi River surveys. 

" It is essentially a semicircular protractor, provided with 




Fig. 66. 



* Designed by J. A. Ockerson and described in Report of the Miss. River 
Commission for 1885. Manufactured by A. S. Aloe, St. Louis, Mo. 
17 



258 SURVEYING. 



a needle-pointed pivot at its centre, and having the straight 
edge graduated so that distances can be measured off each 
way from the pivot ; the angular deflection is given by the 
graduated circle, reading from a point marked on the paper. 
The bottom of the plate is flush with the bottom of the pro- 
tractor, and the hole F is at the centre, and should be only 
large enough to admit a fine needle. The screw D has a hole 
drilled in its axis to admit the needle-point. It is also split, 
so that when it is screwed down it will clamp the needle 
firmly. If the latter is broken, it can readily be replaced by a 
new one. In addition to the scale on the beveled edge, a 
diagonal scale is also provided as shown. This instrument 
combines all the requisites for rapid and accurate plotting of 
points located by polar co-ordinates or by intersections. 

In using this protractor the needle-point is placed at, say, 
the first station, and pressed firmly down. A meridian line is 
then decided upon, and a point is marked on it at the outer 
edge of the protractor circle. This will be the initial point 
from which the angles will be read. As azimuth is read 
from the south around by the west, it is plain that the circle, 
numbered as shown and revolved about the pivot till the 
proper reading coincides with the meridian line, will give the 
direction of the required point along the graduated diameter, 
while from the latter the distance can be pricked off. A point 
can be plotted in any direction without lifting the protractor 
from its position. 

In going to the second station it is not necessary to draw 
a meridian line through it. The azimuth between the first 
and second stakes being known, if the pivot be set at the lat- 
ter, and the protractor revolved so that the straight edge coin- 
cides with the line passing through the two stakes, then the 
point on the circle corresponding to the azimuth of the line 
will be a point on the meridian line. This point being marked 
on the paper is the origin for the angles plotted from the 



TOPOGRAPHICAL SURVEYING. 259 



second station, and it is evident that they will bear the proper 
relations to the points plotted from the first station. 

Other methods are employed for plotting the side shots, 
such as solid half-circle protractors, of paper or horn, weighted 
in position, with their centres over the station. This is ori- 
ented on a meridian drawn through the point, and then all the 
points plotted whose azimuth falls between o° and 180 , when 
the protractor is laid over on the other side, and the remaining 
points plotted. In this case the ruler is laid across the pro- 
tractor, with some even division at the station. This method 
is more troublesome, less rapid, and defaces the drawing more, 
than the other methods given above. The plotter should have 
an assistant to read off to him from the note-book. When 
all the elevations have been plotted, the contour lines are 
sketched in. 

The plotting should keep pace with the field-work as close- 
ly as possible, being done at night and at other times when the 
field-work is prevented or delayed. In difficult ground the 
map could be carried into the field and the contours sketched 
in on the ground. At least the stadia lines should be plotted 
up and checked before the observer leaves the immediate local- 
ity. Where the elevations are checked on B.M.'s, these checks 
should be immediately worked out. This much, at least, could 
be done each evening for that day's work. 

215. Contour Lines. — In engineering drawings the config- 
uration of the surface is represented by means of contour lines. 
A contour line is the projection upon the plane of the paper of 
the intersection of a horizontal, or rather level, plane with the 
surface of the ground. These cutting level planes are taken, 
five, ten, twenty, fifty, or one hundred feet apart vertically, 
beginning with the datum-plane, which is usually taken below 
any point in the surface of the region. Mean sea-level is the 
universal world's datum which should always be used when 
a reasonably accurate connection with the sea can be ob- 



260 SURVEYING. 



tained.* Such contour lines are shown on Plate II. The proper 
drawing of these contours requires some accurate knowledge of 
the surface to be depicted, aside from the elevations of isolated 
points plotted on the map. This knowledge may consist of a 
vivid mental picture of the ground, derived from personal ob- 
servation, or it may be gained from sketches made upon the 
ground. Even with this knowledge the draughtsman must 
keep vividly in mind the true geometrical significance of the 
contour line, in order to properly depict the surface by this 
means. The ability to draw the contour lines accurately on a 
field-sheet is the severest test of a good topographer. They 
are first sketched and adjusted in pencil and then may be 
drawn in ink. 

A few fundamental principles may be stated that will assist 
the young engineer in mastering this art. 

1. All points in one contour line have the same elevation 
above the datum-plane. 

2. Where ground is uniformly sloping the contours must 
be equally spaced, and where it is a plane they are also straight 
and parallel. 

3. Contour lines never intersect or cross each other. 

4. Every contour line must either close upon itself or ex- 
tend continuously across the sheet, disappearing at the limits 
of the drawing. It cannot have an end within these limits (an 
apparent exception, though not really one, is the following). 

5. No contour should ever be drawn directly across a 
stream or ravine. The contour comes to the bank, turns up 
stream, and disappears in the outer stream line. If the bed of 
the stream, or ravine, ever rises above this plane, then the 
contour crosses it ; but in the case of a stream the crossing is 
never actually shown. In the case of a ravine the crossing is 
shown, if points have been established in its bed. 

6. Where a contour closes upon itself, the included area 



* See in Chapter XIV., Precise or Geodesic Levelling. 



TOPOGRAPHICAL SURVEYING. 26 1 

is either a hill-top or a depression without outlet. If the 
latter, it would in general be a pond or lake. In other words, 
such contours enclose either maximum or minimum points of 
the surface. 

7. If a higher elevation seems to be surrounded by lower 
ones on the plot, it is probably a summit ; but if a lower eleva- 
tion seems to be surrounded by higher ones, it is probably a 
a ravine, or else an error ; otherwise it is a depression without 
outlet, in which case there would probably be a pool of water 
shown. 

8. Contour lines cut all lines of steepest declivity, as well 
as all ridge and valley lines, at right angles. 

9. Maximum and minimum ridge and valley contours must 
go in pairs ; that is, no single lower contour line can intervene 
between two higher ones, and no single higher contour line can 
intervene between two lower ones. 

10. Vertical sections, or profiles, corresponding to any line 
across the map, straight or curved, can be constructed from a 
contour map, and conversely a contour map may be drawn 
from the profiles of a sufficient number of lines. 

11. Each contour is designated by its height above the 
datum-plane, as the fifty-foot contour, the sixty-foot contour, 
etc. In flat country, where the contour lines are few and wide 
apart, always put the number of the contour on the higher 
side, otherwise it sometimes may be impossible to tell on which 
side is the higher ground. 

12. In taking surface-elevations for determining contour 
lines, points should always be taken on the ridge and valley 
lines, and at as many intermediate points as may be desirable. 
There are two general systems of selecting these points. By 
one system points are chosen approximately in lines or sec- 
tions cutting the contours about at right angles, the critical 
points being the tops and bottoms of slopes ; while by the 
other system points are selected nearly in the same contour 
line, — that is, on the same horizontal plane, — the critical points 



262 SURVEYING. 



being the ridge and valley points, these being the points of 
maximum and opposite curvature in the contour lines them- 
selves. By the second method one or two principal contours 
may be followed continuously, the points being taken as nearly 
as may be on these contour lines. If such principal contours 
are 50 feet apart, then when these are accurately drawn on the 
map, any desired number of additional contours may be inter- 
polated between the principal ones. 

216. The Final Map. — The field-sheets are drawn as de- 
scribed above, in pencil, or partly in pencil and partly in ink, 
or wholly in ink, according to the use to be made of them. If 
they are simply to serve as the embodiment of the field-sur- 
vey, to be used only for the construction of the final maps, 
they are usually left in pencil, a six-H pencil being used. The 
field-sheets are usually small, about 18x24 inches. The final 
sheets maybe of any desired size. Usually several field-sheets 
are put on one final sheet, which will be worked up wholly in 
ink, or color, the scale remaining the same. The work on the 
field-sheet is then simply transferred to the final sheet by the 
most convenient means available. Tracing-paper (not linen) 
may be used. This is carefully tacked or weighted down over 
the field-sheet, and the principal features, such as triangulation 
stations, stream and contour lines, roads, buildings, fence lines, 
etc., are traced in ink. The tracing-paper is then removed and 
laid upon the final sheet, orienting it by making the triangula- 
tion stations on the tracing coincide with the corresponding 
stations on the final sheet, where they have been carefully 
plotted from the triangulation reduction. All the matter on 
the tracing may now be transferred to the paper beneath by 
passing over the inked lines with a dull point, bearing down 
hard enough to leave an impression on the paper below. If 
preferred, the tracing may have its under surface covered with 
plumbago (soft pencil-scrapings), after the tracing is made, and 
then with a very gentle pressure of the tracing-point will leave 
a light pencil line on the final sheet. " In either case, when the 



TOPOGRAPHICAL SUR VE YING. 263 

tracing is removed, these lines may be inked in on the final 
sheet. 

If the map is to be photo-lithographed it must be drawn 
wholly in black, as given in Plates II. and III. If not, it is best 
to use some color in its execution. The water-lines may be 
drawn in blue, and the contours in brown on arable land, and in 
black on barren or rocky land. In this way the character of the 
surface may be partly given. Where the slopes are very steep 
the contour lines become nearly coincident, but to further em- 
phasize the uneven character of the ground, cross-hatching, or 
hachures, may be employed on slopes greater than 45 from 
the horizontal. All these conventional practices are illustrated 
on Plate III., except the use of colors, this map having been 
drawn for the purpose of being photo-lithographed. Plate II. is 
a photo-lithograph copy of a student's map of the annual field 
survey of the engineering students of Washington University. 

217. Topographical Symbols are more or less conven- 
tional, and for that reason given forms should be agreed upon. 
The forms given in Plate III. were used on all the Mississippi 
River surveys made under the Commission, and are recom- 
mended as being elegant and fairly representative or natural. 
Evidently the rice, cotton, sugar, and wild-cane symbols would 
find no place in maps of higher latitudes. The cypress-tree 
symbols may be used for pine to distinguish them from decid- 
uous growth, and the sugar-cane symbol could be used for 
corn if desired. It is not important to distinguish between 
different kinds of cultivated crops, since these are apt to change 
from year to year, but it is sometimes desirable to do so to 
give a more varied and pleasing appearance to the map. The 
grouping of the trees in a large forest is also varied simply for 
the appearance, to prevent monotony. Colors are sometimes 
used in place of pen-drawn symbols, but these are necessarily 
so very conventional as to require a key to interpret them, and 
besides it makes the map look cheap and unprofessional. 

218. Accuracy of the Stadia Method. — In measuring dis- 



264 SURVEYING. 



tances by stadia the errors made in reading the rod are as apt 
to be plus as minus. They therefore follow the law of com- 
pensating errors, which is that the square root of the number 
of errors remains (probably) uncompensated. If the rod was 
properly graduated, therefore, the only error is that from read- 
ing the position of the wires. On inclined sights the distance 
read on the rod is accurately reduced to the horizontal by 
means of proper tables or diagrams. There is another pecu- 
liarity of this system, and that is that the accuracy depends 
very largely on the state of the atmosphere. If this is clear 
and steady the accuracy attainable for given lengths of sight 
is much greater than when it is either hazy or very unsteady 
from the effects of heat. Or, for a give'n degree of accuracy, 
the lengths of sight may be taken much longer under favorable 
atmospheric conditions than under unfavorable ones. It is 
impossible, therefore, to specify any given degree of precision 
for given lengths of sight for all atmospheric conditions. The 
results obtained on the U. S. Lake Survey are perhaps a 
fair average for various conditions. On that service the errors 
of closure of 141 meandered lines was computed with a mean 
result of one in 650. The lengths of sight averaged from 
800 to 1000 feet, with a maximum length of about 2000 feet. 
The official limit of error of closure was one in 300. The 
average length of the lines run was one and a half miles. If 
care is taken to shorten up the sights for unsteady atmo- 
sphere, and to reduce all readings to the horizontal, it would 
not be difficult to reduce the error of closure on lines aver- 
aging from one to two miles in length, to one in 1000 or one 
in 1200. Since the absolute error increases as the square 
root of the length of the line run, it is evident that the relative 
error diminishes as the length of line increases. Thus, for a 
single reading of say 400 feet the error might possibly be two 
feet, but for 100 such sights the error probably would be but 
10 X 2 feet = 20 feet, the distance now being 40,000 feet, 
giving an error of one in 2000. 



CHAPTER IX. 
RAILROAD SURVEYING. 

219. Objects of the Survey. — Since the transit and stadia 
are the best means of making a topographical survey, so they 
are the means that are best adapted to make a railroad survey, 
so far as this is a topographical survey. 

The map of a railroad survey may serve two purposes : 

First, to enable the engineer to make a better location of 
the line than could be done in the field. 

Second, to give all necessary data relating to right of way, 
as the drawing of deeds, assessment of damages, etc. 

In flat or gently undulating country, it may not be advis- 
able to locate by a map ; but even here the map is quite as 
essential for determining questions relating to the right of way. 

In either case, therefore, a good topographical map of the 
line is of prime importance, and all the data for this map may 
be taken on the preliminary survey.* 

Both these ends may be served by the same map. The 
method of location by contours (sometimes called " paper lo- 
cation") is often absolutely necessary in rough ground, but is 
still more often judicious in simpler work, inasmuch as a better 
location can often be made in this way. 

220. The Field-work. — In this case there would be no 
A's or B.M.'s to check on; but the errors in distance and ele- 
vation would be no more, probably, than are now made on 

* By " preliminary survey" is here meant a survey of a belt of country 
which it is expected will embrace the final line, and not a mere reconnoisance 
made to determine the feasibility of a line, or which of several lines is the best. 






266 SURVEYING. 



preliminary surveys. In fact, the errors in distance would not 
be nearly so great, unless the chain be tested frequently for 
length, and the greatest care taken on irregular ground. If a 
chain ICO feet long has 600 wearing-surfaces, which most of 
them have, and if each of these surfaces be supposed to wear 
O.Oi inch, which it will do in the course of a 200- or 300-mile 
survey, then the chain has lengthened by six inches, or the 
error in distance is now 1 in 200 from this cause alone. If we 
add to this the uncertain errors that come from chaining up 
and down hill, and over obstructed ground, it is certain that 
the stadia measures will be much the more accurate. 

In the matter of elevations, since the local change of ele- 
vation is alone significant, and not the total difference of ele- 
vation of points at long distances apart, the line of levels 
carried by the stadia would be amply sufficient for a prelimi- 
nary survey. 

The following observations are applicable to the prelimi- 
nary survey for final location, when it is expected the line will 
be included in the belt of country surveyed : 

1st. All data should be taken that will contribute to the so- 
lution of all questions of location, such as elevations for con- 
tour lines ; streams requiring culverts, trestles, or bridges, and 
the necessary size of each, if possible ; all depressions which 
cross the line, and will require a water-way, together with the 
approximate size of the area drained ; highways and private 
roads or lanes ; buildings of all kinds, fences, and hedges ; 
character of surface, as rock, clay, sand, etc. ; character of 
vegetation, as cultivated, forest, prairie, marsh, etc. ; the loca- 
tion of any natural rock that may be used for structures on the 
line, such as culverts or abutments ; high-water marks if in a 
bottom subject to overflow; and, in fact, all information which 
will probably prove of value in determining the location, or in 
making up a report with estimates to the board of directors, or 
in letting contracts for earthwork. 



RAILROAD SURVEYING. 267 

2d. All data that may be found useful in respect to land 
titles or right of way, or that may relate to claims for dam- 
ages, such as section corners, boundaries, fences, buildings, 
streets, roads, lanes, farm roads, cultivated and uncultivated 
land, as well as such as may be cultivated, public and private 
grounds, orchards, forests, together with the value of the forest 
timber, mineral lands, stone quarries, proximity to villages, 
etc. Since the bearings and position of all boundary-lines are 
of great importance in the matter of right of way, every such 
boundary should have at least two readings upon it in the 
field ; and these should be as far apart as possible. 

221. The Maps. — Before any plotting is done, two ques- 
tions of importance must be decided. They are — first, 
whether one set of maps is to serve for both the location and 
for the further use of the company, or whether a set of contour 
maps, worked up in pencil, shall serve for the location, and 
another set for the continuous use of the company ; second, 
what shall be the scale of the maps ? These will be argued 
separately. 

Whether one or two sets of maps will be decided on, will de- 
pend largely on the care that is exercised with the locating- 
sheets. If these are carefully worked up for the location, and 
kept clean, they can be utilized for the final maps. If they 
become too badly soiled by field use, new sheets would prob- 
ably be substituted for the uses of the company. 

If it is expected, at the start, to have a different set of 
sheets for the final maps, then " protractor sheets" should be 
used for the location. In this case, plot on these sheets only 
such of the field-notes as will contribute to the location ; and 
these need only be plotted in pencil. When the location has 
been made, such features may be transferred from the locating- 
sheets to the final maps, as may be desired. These would con- 
sist mainly in the stadia stations, the contours, and the located 
line. The rest of the field-notes may then be plotted on the 
final sheets, and the whole worked up in ink. 



268 SURVEYING. 



If, on the other hand, one set of maps is to serve both pur- 
poses, then it would, perhaps, be best to use plain sheets, as 
the protractor circle would somewhat disfigure the final maps. 
The protractor sheets would, however, furnish a ready means 
of taking off the bearings of lines from the final charts, which 
might be thought to compensate for the slight marring of the 
map's appearance. If plain sheets are chosen, then they should 
be divided into squares by lines drawn in ink parallel to the 
sides of the paper, in the direction of the cardinal points of 
the compass. Both the stadia stations and the side-readings 
may then be plotted by means of the auxiliary protractor, this 
being oriented by the meridian lines on the sheet. Even here, 
only those readings would at first be plotted that will contrib- 
ute to the location, and these marked in pencil. After the 
location has been decided on, and the location notes taken off, 
as described below, then the stadia stations, contour lines, the 
located line of road, and such other features as should be pre- 
served on the final map, are inked in, and the map thoroughly 
cleaned. The rest of the field-notes may now be plotted, and 
the map finished up. 

If the road runs through a settled region, the questions of 
right of way are among the first things to be settled ; so that 
preliminary maps showing the relation of the road belt to the 
property lines are essential to the settlement of damages, and 
to obtaining the right of way from the property-holders. 
Coincident, therefore, with the making of maps to determine 
the location must come the construction of preliminary right- 
of-way maps or tracings. On these latter need be plotted only 
the boundary-lines, fences, more important buildings, roads, 
etc., or just sufficient to enable the right-of-way agent to nego- 
tiate intelligibly with the property-owners.* Neither the lo- 

* For an excellent article on the subject of right-of-way maps and permanent 
railway-property records, by Charles Paine, see The Railroad Gazette of Nov. 
14, 1884. 



RAILROAD SURVEYING. * 269 

eating nor the final map should be on a continuous roll. The 
roll requires more room for storage, is more apt to get dusty, 
and is much more inconvenient for reference. When sheets 
are used, the survey plot covers a more or less narrow belt 
across the map. One of the edges of the sheet, either where 
the plot enters upon it or disappears from it, should be trimmed 
straight, and the plot extended quite to this edge. This edge 
is then made to coincide with one of the parallel or meridian 
lines of the next sheet ; so that when the line is plotted, the 
sheets may be tacked down in such a way as to show the con- 
tinuous plot of the survey. 

The scale of the map will depend on whether or not separate 
sets of charts are to serve the purposes of location and of the 
continuous use of the company. For the purpose of location, 
a scale of 400 feet to one inch does very well ; but for the final 
detail sheets the scale should be larger. If both purposes are 
to be served by one set of maps, then the scale should be 
about 200 feet to one inch,* with 5- or 10-foot contours. The 
. sheets should be about twenty by twenty-four inches. 

222. Plotting the Survey. — In case the map is plotted on 
a protractor sheet, the methods of plotting will be identical 
with those for general topographical work, except that here 
there will be no checks, either for distance, azimuth, or eleva- 
tion, except such as are carried along or independently de- 
termined. For distance, there is no check, except the dupli- 
cate readings between instrument stations, unless the survey 
is through a region which has already been surveyed. In this 
case the section lines may serve as a check on the distances. 

The azimuth should be checked at every station by reading 
the needle, as described on p. 248, and also by independently 
determining the meridian frequently, either by a solar attach- 
ment or by a stellar observation. If the line is not nearly 



* Some engineers prefer a scale ot 100 leet to one inch for the final charts of 
the company. 



270 SURVEYING. 



north and south, or, in other words, if it is extended materially 
in longitude, then the azimuth must be constantly corrected 
for convergence of meridians, as is shown in Chap. XIV. 

The elevations can only be checked by the duplicate read- 
ings between instrument stations.* All the greater care 
should be used, therefore, on readings between stations. 

The first plotting, whether there are to be two sets of maps 
or one, will consist in representing on the sheet only such data 
as will assist in deciding on the location. These will be mainly 
contour points, streams, important buildings near the line, 
principal highways, other lines of railway, villages with their 
streets and alleys near the proposed location, the lines of de- 
markation between cultivated and timbered or wild land, etc. 
From the plotted elevations, aided by the sketches in the note- 
book, the contour lines are drawn in ; if necessary, this may 
be done on the ground. This is sufficient for determining 
upon a location. 

When this has been done, then the natural features, the 
contour lines, the stadia stations, and the located line, may be 
inked in (or transferred by means of tracing-paper, in case the 
final maps are to be on separate sheets), and the remainder of 
the notes plotted. 

In drawing the contour lines in ink, make those upon bar- 
ren or rocky land in black, and those on arable land in brown. 
If they are ten feet apart, make every tenth one very heavy, 
and every fifth one somewhat heavier than the others. If this 
be done, only the 50- and 100-foot contours need be numbered. 
In case a map does not contain at least two of these numbered 
contours, then every contour which does appear on the map 
should be numbered, giving its elevation above the datum of 
the survey. 

* It may be observed that the same lack of sufficient checks on the distance, 
azimuth, and elevation obtains with the ordinary preliminary survey with tran- 
sit, level, and chain. 



RAILROAD SURVEYING. 27 1 

The streams should be water-lined in blue, and an arrow 
should tell the direction of its flow. The name should also be 
given when possible. 

All fences should be shown, and especial pains taken to 
represent division fences in their true position ; for it is from 
this map that the deeds for the right of way are to be drawn. 

Outhouses may be distinguished from dwellings by diago- 
nal lines intersecting, and extending slightly beyond the out- 
line. The character of the buildings may be shown by colors, 
as red for brick, yellow for frame, pale sepia for stone ; the 
outlines always being in black. 

The stadia stations should be left on the finished sheets ; 
3.?,, m case of a disputed boundary, or for other cause, the map 
may be replotted if the positions of the instrument stations 
are left on it. The numbers of the stations should, of course, 
be appended. 

The magnetic bearings of boundary-lines may be given on 
the map, or they may be determined, as occasion requires, by 
means of the auxiliary protractor and the true meridian lines 
when the variation of the needle is known. For this purpose, 
the magnetic meridian should be drawn on each map, diverg- 
ing from one of the meridian lines, and the amount of the 
variation marked in degrees and minutes. 

223. Making the Location. — When a preliminary survey 
is made, as above described, for the purpose of making what 
is called a " paper location," the location is first made on the 
map, and then staked out in the field. 

Every railroad line is a combination of curves, tangents, 
and grades ; and it is the proper combination of these which 
makes a good location. If it be assumed that the line is to be 
included in the belt of country surveyed, then the map con- 
tains all the data necessary to enable the engineer to select the 
best arrangements of curves, tangents, and grades it is possible 
for him to obtain on this ground. This selection can be made 



272 SURVEYING. 



with much more certainty than is possible on the ground, 
where the view is generally obstructed, and where grades are 
so deceptive. 

It is no part of this treatise to discuss the various problems 
that enter into the question of a location, but only to show 
how to proceed to make a location that may satisfy any given 
set of conditions, by means of the contour map. 

The contours themselves will enable the engineer to decide 
what the approximate grades will have to be. Suppose a grade 
of 0.5 foot in 100 feet, or 26.4 feet to the mile, has been fixed 
upon. It is now known that the line should follow the gene- 
ral course of the contours, except that it should cross a 10-foot 
contour every 2000 feet. Spread the dividers to this distance, 
taken to scale, and mark off in a rough way these 2000-foot 
distances as far as this grade is to extend ; and do the same 
for the successive grades along the line. Knowing the grade 
of the line at the beginning of the sheet, the problem is to ex- 
tend this line over the sheet so as to give the best location 
one can hope to get on this ground with the available 
means. 

First, starting from the initial fixed point of line on the 
map, sketch in a line which will follow the contours exactly, 
crossing them, however, at such a rate as to give the necessary 
grade. This is the cheapest line, so far as cut and fill are con- 
cerned. Of course, where depressions or ridges are to be 
crossed, the line must cross over from a given contour on one 
side to the corresponding contour on the other, and then fol- 
low along the contour again. 

Second, mark out a series of tangents and curves which will 
follow this sketched line as nearly as it is possible for a rail- 
road to follow it. This will not be the final location, but it is 
valuable for study. This line will be faulty from having too 
many and too sharp curves, and too little tangent. 

Third, draw in a third line, as straight as possible, and with 



RAILROAD SURVEYING. 



273 



as low grade of curves as possible consistent with a reasonable 
amount of earthwork and a proper distribution of the same. 

For the purpose of deciding what degree of curve is best 
suited to the ground for«a given deflection-angle, it is well to 
have a series of paper templets made, with the various curves 
for their outer and inner edges. Of course, these are cut with 
radii laid off to the scale of the drawing. It is still more con- 
venient to have these curves, laid off to scale, on a piece of 
isinglass, horn, or tracing-paper (not linen), so that this can be 
laid upon the map, and the curve at once selected which will 
follow the contours most economically. Fig. 66 shows such a 
series of curves drawn to a scale of 1600 feet to the inch. 




Fig. 66. 



In this way the line is laid out over the map. The ques- 
tions of greater or less curvature have been balanced against a 
less or greater first cost, and greater or less operating expense. 
The question of shifting it laterally has also been examined, 
and finally a definite location fixed upon which seems to answer 
best to the case in hand. When this is done, it only remains 
to make up the location notes from which the line is to be 
staked out. 
18 



274 



SURVEYING. 



The following is considered a good form for the location 
notes: 



Location Notes for ABC Railroad. From Map No. 



Line. 


Azimuth and 

Deflection 

Angles. 


Length. 


Station. 


Remarks. 


T 


260 40' 


ft. 
1020 


IO-f-20 




3° CR. 


+ 18° 30' 


617 


16 + 37 




T 


279° IO' 


2670 


43+7 




4° C.L. 


- 12° 20' 


308 


46+15 




T 


266 50' 


680 


52 + 95 





The first column designates the tangents and curves, and 
gives the degree of the curve, and the direction of its curva- 
ture, whether right or left. If it curve toward the right, the 
azimuth of the next tangent will be increased, and hence its 
sign is plus, and vice versa. 

The second column gives the azimuths of the tangents and 
the deflection-angles of the curves. Each azimuth is seen to 
be the algebraic sum of the two preceding angles. 

The 1 1 > hird column gives the lengths of the tangents as meas- 
used from the map, and the lengths of the curves as determined 
by dividing the deflection-angle by the degree of the curve. 
Thus, 12 20' = 12°. 33, and 12°. 33 -+-4= 308, which is the 
length of the curve in feet.* 

The fourth column gives the stations and pluses for the 
P.C.'s and the P.T.'s. These quantities are simply the con- 
tinued sum of those in the third column. 

The first, second, and fourth columns now give all the infor- 

* It is a great convenience to have at least one vernier, in railroad work, 
graduated to read to hundredths of a degree. The case here given is only one 
of many similar cases; but the principal advantage is in running the fractional 
parts of curves when the curve chosen is some even degree, as here taken. 






RAILROAD SURVEYING. 275 

mation necessary to stake out the line. The stadia is no longer 
to be used, but a transit and chain, as is ordinarily done. 

The tangents need not be run out to their intersection ; but 
when the P.C. is reached, according to the location notes taken 
from the map, set up the instrument, and stake out the curve 
as far as possible, or around to the P.T. In either case, when 
the instrument is to be moved, make a note of the forward 
azimuth, and go forward and orient on the last station the 
same as when moving between two H's. If the instrument be 
moved to the P.T. direct, then, after orienting back on the 
P.C, turn off to the azimuth given for the next tangent, and 
go ahead. The tangents could be run out to the intersection 
and the point occupied by the instrument; for a check, if 
thought desirable. The telescope is never reversed in laying out 
the line from the system of notes above given. 

With careful work, the line ought thus to be run out, and 
the curves put in at once. We have supposed there was no 
regular line cleared out on the preliminary, so the necessary 
clearing would all have to be done on the location. 

A levelling party follows the transit, and obtains the data 
for constructing a profile and for determining the exact grades. 

The stadia has served its purpose when it has enabled the 
engineer to select the most favorable position for the line. 
The transit, chain, and level must do the balance. It is not 
improbable that occasional modifications will be introduced in 
the field, even though the survey and the location have been 
made with the greatest possible care. 

224. Another Method of making the preliminary survey 
from which to determine the final location is as follows : 

Run a transit and chain line, setting 100-foot stakes, as 
nearly on the line of the road as can be determined by eye. 
Follow this party by a level party which obtains the profile of 
the transit line. A third party of one or more topographers 
takes cross-sections at each 100-foot stake by means of a 



276 SURVEYING. 



pocket-compass, chronometer, and hand-level. These cross- 
sections show the ground on either side of the line as far as 
desirable by slope and distance, these latter being either meas- 
ured by tape or paced. It is evident that contour lines could 
be worked out from these data, but these would not be needed 
if the distances and slopes were well determined, since these 
give a better cross-section than contours alone could do. 

The objections to this method are in the poor means it fur- 
nishes for accurate determination of either distances or slopes, 
and the haste with which it is usually «done. There can be no 
question but that accurate distances and slopes on cross-sections 
IOO feet apart would give fuller data than even five-foot con- 
tours accurately drawn. But to be accurately determined the 
slope would have to change at all points — in other words, it 
would be a curve. As to whether the slopes and distances as 
they would probably be taken would give a better idea of the 
ground than five-foot contours determined by the stadia 
method, and the relative cost of the two systems, are matters 
of experience. Both systems are competent to give a good 
location when they are well executed. 

Note. — The further study of railroad surveying falls within the province of 
the various railroad field-books, which are printed in pocket form and contain 
the necessary tables for laying out a line of road. Having learned the con- 
struction and use of surveying-instruments, and the general methods of topo- 
graphical surveying and levelling, the special applications to railroad location 
given in the field-books are readily mastered. They will therefore not be 
further considered in this work. 



CHAPTER X. 
HYDROGRAPHIC SURVEYING. 

225. Hydrographic Surveying includes all surveys, for 
whatever purpose, which are made on, or are concerned with, 
any body of still or running water. Some of the objects of such 
surveys are the determination of depths for mapping and navi- 
gation purposes ; the determination of areas of cross-sections, 
the mean velocities of the water across such sections, and the 
slope of the water surface ; the location of buoys, rocks, lights, 
signals, etc. ; the location of channels, the directions and ve- 
locities of currents, and the determination of the changes in 
the same ; the determination of the quantity of sediment car- 
ried in suspension, of the volume of the scour or fill on the 
bottom, or of the material removed by artificial means, as by 
dredging. 

A hydrographic survey is usually connected with an ex- 
tended body of water, as ocean coasts, harbors, lakes, or riv- 
ers. The fixed points of reference for the survey are usually 
on shore, but sometimes buoys are anchored off the shore and 
used as points of reference. All such points should be accu- 
rately located by triangulation from some measured base 
whose azimuth has been found. The buoys will swing at 
their moorings within small circles, these being larger at low 
tide than at high, but the errors in their positions should never 
be sufficient to cause appreciable error in the plotted positions 
of the soundings. Where soundings need to be located with 
great exactness, buoys could not be relied on. The triangula- 
tion work for the location of the fixed points of reference dif- 
fers in no sense from that for a topographical survey. In fact, 



278, SURVEYING. 



a hydrographic survey is usually connected with a topographical 
survey of the adjacent shores or banks, the triangulation 
scheme serving both purposes. It is not uncommon, however, 
to make a hydrographic survey for navigation purposes sim- 
ply, wherein only the shore-line and certain very prominent 
features of the adjacent land are located and plotted. This is 
the practice of the U. S. Hydrographic Office in surveying for- 
eign coasts and harbors. In this case the work consists almost 
wholly in making and locating soundings for a certain limiting 
depth, as one hundred fathoms, or one hundred feet, inward 
to the shore, and along the coast as far as desired. The length 
and azimuth of a base-line are determined and the latitude ob- 
served by methods given in Chapter XIV. The longitude is 
found by observing for local time, and comparing it with the 
chronometer time which has been brought from some statio.11 
whose longitude was known. Whenever telegraphic com- 
munication can be obtained with a place of known longitude, 
the difference between the local times of the two places is 
found by exchanging chronographic signals. No special de- 
scription will be here given of the methods used in this part of 
the work, as they are all fully described in Chapter XIV. 



THE LOCATION OF SOUNDINGS. 

226. Methods. — The location of a sounding can be found 
with reference to visible known points by (1) two angles read 
at fixed points on shore ; (2) by two angles read in the boat; 
(3) by taking the sounding on a certain range, or known line, 
and reading one angle either on shore or in the boat ; (4) 
by sounding along a known range, or line, taking the soundings 
at known intervals of time, and rowing at a uniform rate ; (5) 
by taking the soundings at the intersections of fixed range 
lines ; (6) by means of cords or wires stretched between fixed 



HYDROGRAPHIC SURVEYING. 279 

stations, these having tags, or marks, where the soundings are 
to be taken. These methods are severally adapted to differ- 
ent conditions and objects, and will be described in order. 

227. Two Angles read on Shore. — If two instruments 
(transits or sextants) be placed at two known points on shore, 
and the angles subtended by some other fixed point, and the boat 
be read by both instruments, when a sounding is taken, the in- 
tersection of the two pointings to the boat, when plotted on the 
chart containing the points of observation duly plotted, will 
be the plotted position of the sounding. If three instruments 
are read from as many known stations, then the three point- 
ings to the boat should intersect in a point when plotted, thus 
furnishing a check on the observations. The objections to 
this method are that it requires at least two observers, and 
these must be transferred at intervals, as the work proceeds, in 
order to maintain good intersections, or in order to see the 
boat at all times. While an observer is shifting his posi- 
tion the work must be suspended. If there are long lines of 
off-shore soundings to be made and there are no fixed points or 
stations on shore of sufficient distinctness or prominence to be 
observed by the sextant from the boat, then this method must 
be used. When the angles are read on shore signals should be 
given preparatory to taking a sounding, and also when the 
sounding is made. If, however, the soundings are taken at 
regular intervals the preparatory signal may be omitted, and 
only the signal given when the sounding is taken. This 
usually consists in showing a flag. The instrument may be set 
to read zero when pointing to the fixed station. This reading 
need only be taken at intervals to test the stability of the 
instrument. 

228. By Two Angles read in the Boat to three points on 
shore whose relative positions are known. This is called the 
"three-point" problem. Let A, C, and B be the three shore 
points, being defined by the two distances a and b and the angle 



280 



SURVEYING. 



C. Let the two angles P and F be measured at the point P. 
The problem is to find the distances AP and BP. 

{a) Analytical Solution. — Let the un- 
known angle at A be x, and that at B be 
b y. Then we may form two equations 
from which x and y may be found. 
For, 



a sin x b sin y 
sin P sin P v ' 



Also, *+ j = 3 6o° -(/>+/* +<;) = ;?. . . .. (2) 




From (2), y — R — x, 



and 



sin y = sin R cos ;tr — cos R sin ;tr. 



Substitute this value of sin y in (1), reduce, and find 



cot x = 



a sin P -f- b sin P cos R 
b sin Psin R 



cot .£ 



# sin P' 



\b sin P cos R 



+ 1 (3) 



When # and j are found, the sides AP and i?P are readily 
obtained. This is perhaps the simplest analytical solution of 
the problem. 

{b) Geometrical Solution. — The following geometrical solu- 
tion is of some interest, though it is seldom used : 

Let A, C, and B be the fixed points as before, and Pand 
F the observed angles. Having the points^, B, and C plotted 



HYDROGRAPHIC SURVEYING. 



28l 




Fig. 69. 



in their true relative positions, draw from A the line AD, 
making with AB the angle F (CPE), and from B the line BD, 
making with AB the angle P (APC), 
cutting the former line in D. Through 
A, D, and B pass a circle, and through C 
and D draw a line cutting the circum- 
ference again in P. The point P is the 
plotted position of the point of observa- 
tion from which the angles P and P' were 
measured. 

For P must lie in the circumference 
through ADB by construction, otherwise 
ABD would not be equal to APD, as they 
are both measured by the same arc AD. The same holds for 
the angle P' '. Also, the line PD must pass through C, other- 
wise the angle APC would be greater or less than P, which 
cannot be. The point P is therefore on the line CD, and also 
on the circumference of the circle through ADB, whence it is 
at their intersection. 

This demonstration is valuable as showing when this 
method of location fails to locate, and when the location is 
poor. For the nearer the point D comes to C the more un- 
certain becomes the direction of the line CD, and when D falls 
at C — that is, when P is on the circumference of a circle through 
A, B, and C — the solution is impossible, inasmuch as P may 
then be anywhere on that circumference without changing the 
angles P and P'. This is also shown by equation (3), above; 
for if A, C, B, and Pall fall on one circumference, then x -f- y 
= R = 180 ; whence cot x = 00 X o, which is indeterminate. 
For cot R = — 00, and cos R= — 1. Also a sin P' = b sin P, 
both being equal to the perpendicular from C on AB. The 
equation then becomes 



cot x = 00(1 — 1) = 00 X o. 



282 SURVEYING. 



(c) Mechanical Solution. — If the three known stations be 
plotted in position and the two observed angles be carefully 
set on a three-armed protractor,* then when the three radial 
edges coincide with the three stations, the centre of the pro- 
tractor circle corresponds to the position of the point of obser- 
vation. With a good protractor this method gives the posi- 
tion of the point as closely as the nature of the observations 
themselves would warrant. It is the common method of plot 
ting soundings when two sextant angles have been read from 
the sounding boat. 

The goniograph, described on p. 113, is designed to serve 
both for reading the angles and for plotting of point, replacing 
therefore both the sextant and protractor in this work. 

(d) Graphical Solution. — The angles may be laid off on 
tracing-paper or linen by lines of indefinite length, and this 
laid on the plot and shifted in position until the three radial 
lines coincide with the three stations, when their intersection 
marks the point of observation. This is the most ready 
method of plotting such observations when no three-armed 
protractor is available. 

The advantages of this method of locating soundings are 
that it requires but one observer, no time is lost in changing 
stations, and the party are all together, and hence there can be 
no misunderstandings in regard to the work. If the soundings 
are made in running water, so that the boat cannot be stopped 
long enough to read two sextant angles, two sextants are 
sometimes used with one observer, he setting both angles and 
reading them afterwards ; or two observers may be employed 
in the same boat and the angles taken simultaneously. 

229. By one Range and one Angle.— The range may be 
two stations or poles set in line on shore, or it may consist of 
one point on shore and a buoy set at the desired position off* 



* For description, with cut, see p. 167. 



H YDROGRAPHIC SUR VE YING. 



283 




shore. If buoys are used they must be located by triangulation 
from the shore stations. A triangulation system along a rocky 
or wooded coast may consist of one line of sta- 
tions on shore and a corresponding line of buoys. 
The angles are read only from the shore stations, 
two angles in each triangle being observed. If 
the buoys are well set and the work done in calm 
weather, the results will be good enough for to- 
pographical or hydrographical purposes. The 
stations and buoys should be opposite each other, 
as in the figure, and readings taken to the two 
adjacent shore stations and to the three nearest 
buoys from each shore station. If the length of 
any line of this system be known, the rest can be 
found when the angles at A, B, C, and D are A , 
measured. In such a system the measured lines 
should recur as often as possible, ordinary chain- 
ing being sufficient. 

230. Buoys, Buoy-flags, and Range-poles. — A conveni- 
ent buoy for this purpose may be made of any light wood, 
eighteen inches to three feet long in tideless waters, and long 
enough to maintain an erect position in tide-waters. It should 
be from six to ten inches in diameter at top, and taper towards 
the bottom. If the buoy is not too long, a hole may be bored 
through its axis for the flag-pole, which may then project two 
or three feet below the buoy and as high above it as desired. 
The buoy rope is then attached to the bottom end of the pole 
and made of such length as to maintain the pole in a vertical 
position in all stages of the tide. The anchor may be any suffi- 
ciently heavy body, as a rock or cast-iron disk. If 'the buoys 
are liable to become confused on the records, different designs 
may be used in the flags, as various combinations of red, white, 
and blue, all good colors for this purpose. 

The range-poles should be whitewashed so as to show up 



Fig. 70. 



284 SUJi VE YING. 



against the background of the shore. The ranges are desig- 
nated by attaching to the rear range-poles slats (barrel-staves 
would serve) arranged as Roman numerals when read up or 
down the pole. If range-poles are relied on, they must be very 
carefully located and plotted, in order to establish accurately 
a long line of soundings from a very short fixed base. 

The observed angle may be either from the boat or from a 
point on shore. In either case any other range-post of the 
series may be used either for the position of the observer, if 
on shore, or for the other target-point if the angle is read fro 
the boat. 

231. By one Range and Time-intervals. — This is a ver 
common and efficient method, and quite satisfactory wher 
soundings need not be located with the greatest accuracy an 
where there is no current. A boat can be pulled in still wate 
with great uniformity of speed ; and if the soundings be taken 
at known intervals with the ends of the line of soundings fixed, 
the time-intervals will correspond almost exactly with th 
space-intervals. If the ends of the line of soundings are not 
fixed by buoys or sounding-stations on shore, but the line sim 
ply fixed by ranges back from the water's edge, the positions o 
the end soundings may be fixed by angle-readings and the bal 
ance interpolated from the time-intervals. 

232. By means of Intersecting Ranges. — This method 
is only adapted to the case where soundings are to be repeate 
many times at the same places. When the object of the sur- 
vey is to study the changes occurring as to scour or fill on th 
bottom it is very essential that the successive soundings shoul 
coincide in position, otherwise discrepant results would prove 
nothing. *Such surveys are common on navigable rivers and 
in harbors. Many systems of such ranges could be described, 
but the ingenious engineer will be able to devise a system 
adapted to the case in hand. 

2 33* By means of Cords or Wires. — In the case of a fixed 



HYDROGRAPHIC SUR VE YING. 



285 



but narrow navigable channel, having an irregular bottom, or 
undergoing improvement by dredging, it may be found advis- 
able to set and locate stakes on opposite sides of the channel, 
to stretch a graduated cord or wire between them, and to locate 
the soundings by this. By such means the location would be 
the most accurate possible. 



MAKING THE SOUNDINGS. 

234. The Lead is usually made of lead, and should be long 
and slender to diminish the resistance of the>water. It should 
weigh from five pounds for shallow, still water, to twenty 
pounds for deep running water, as in large rivers. If depth 
only is required, the lead may be a simple cylinder something 
like a sash-weight for windows. If specimens of the bottom 
are to be brought to the surface at each sounding, the form 
shown in Fig. 71 may be used to advantage. An iron stem, 
/, is made with a cup, c, at its lower end. The 
stem has spurs cut upon it, or cross-bars attached 
to it, and on this is moulded the lead which gives 
the requisite weight. Between the cup and the 
lead is a leather cover sliding freely on the shank 
and fitting tightly to the upper edges of the cup. 
When the cup strikes the bottom, it sinks far 
enough to obtain a specimen of the same, which 
is then safely brought to the surface, the leather 
cover protecting the contents of the cup from be- 
ing washed out in raising the lead. A conical cav- 
ity in the lower end of the lead, lined with tallow, 
is often used, and it is found very efficient for in- 
dicating sand and mud. It is often very essential to know 
whether the bottom is composed of gravel, coarse or fine, sand, 
mud, clay, hard-pan, or rock, and this knowledge can be ob- 
tained with the cup device described above. 

235. The Line should be of a size suited to the weight of 




Fig. 71. 



286 SURVEYING. 



the lead, and made of Italian hemp. It is prepared for use by 
first stretching it sufficiently to prevent further elongation in 
use after it is graduated. Probably the best way to stretch a 
line is to wind it tightly about a smooth-barked tree, securely 
fasten both ends, wet it thoroughly, and leave it to dry. Thei 
rewind as before, taking up the slack from the first stretching, 
and repeat the operation until the slack becomes inappreciable. 
It may now be graduated and tagged. Sometimes it is fastened 
to two trees and stretched by means of a "Spanish windlass/* 
and then wet. It is quite possible to stretch the line too much, 
for sometimes sounding-lines have shortened in use after being 
stretched by this method. Soundings at sea are taken in fath- 
oms. On the U. S. Lake Survey all depths over twenty- 
four feet (four fathoms) were given in fathoms, and all 
depths less than that limit were given in feet. On river an< 
harbor surveys it is common to give depths in feet. Channel- 
soundings on the Western rivers made by boatmen are givei 
in feet up to ten feet, then they are given in fathoms and quar- 
ters, the calls being " quarter-less-twain," " mark-twain," " quar- 
ter-twain," "half-twain," " quarter-less-three, " " mark-three," 
etc., for depths of if, 2, 2^, 2\, 2f , 3, etc., fathoms respectively. 
If the line is graduated in feet leather tags are used even 
five feet, the intermediate foot-marks being cotton or woollei 
strips. The ten-foot tags are notched with one, two, three, 
etc., notches for the 10-, 20-, 30-, etc., foot points, up to fift] 
feet. The fifty-foot tag may have a hole in it, and the 60-, 70- 
80-, etc., foot-marks have tags all with one hole and with one, 
two, three, etc., notches. The intermediate five-foot point* 
have a simple leather tag unmarked. Sometimes the figure* 
are branded on the leather tags, but notches are more easih 
read. The zero of the graduation is the bottom of the lead. 
The leather tags are fastened into the strands of the line ; the 
cloth strips may be tied on. The line should be frequently 
tested, and if it changes materially a table of corrections 



HYDROGRAPHIC SURVEYING. 2%J 

should be made out and all soundings corrected for erroneous 
length of line. 

236. Sounding-poles should be used when the depth is 
less than about fifteen feet. The pole may be graduated to 
feet simply, or to feet and tenths, according to the accuracy 
required. 

237. Making Soundings in Running Water. — The 
sounding-boat should be of the " cutter" pattern, with a sort 
of platform in the bow for the leadsman to stand on. If the 
current is swift, six oarsmen will be required and two ob- 
servers and one recorder. One of the observers may act as 
steersman. If the depth is not more than sixty or eighty feet, 
the soundings are made without checking the boat, the leads- 
man casting the lead far enough forward to enable it to reach 
bottom by the time the line comes vertical. When the depth 
and the current are such as to make this impossible, the boat 
is allowed to drift down with the current and soundings taken 
at intervals without drawing up the lead. The boat is then 
pulled back up stream and dropped down again on another 
line, and so on. 

In still water a smaller crew and outfit may be used, as the 
boat may be stopped for each sounding if necessary. 

The record should give the date, names of observers, general 
locality, number or other designation of line sounded, the 
time, the two angles, the stations sighted, and the depth for 
each sounding, and the errors of the graduated lengths on the 
sounding-line. 

238. The Water-surface Plane of Reference. — In order 
to refer the bottom elevations to the general datum-plane of 
the survey, it is necessary to know the elevation of the water- 
surface at all times when soundings are taken. In tidal waters 
the elevation of " mean tide" is the plane of reference for both 
the topographical and hydrographical surveys, and then the 
state of the tide must be known with reference to mean tide. 



288 SURVEYING. 



This is found from the hourly readings of a tide-gauge (pro- 
vided it is not automatic), the elevation of the zero of which, 
with reference to mean tide-water, has been determined. All 
soundings must then be reduced to what they would have been 
if made at mean tide before they are plotted. 

If the soundings are made in lakes, the datum is usually 
the lowest water-stage on record ; and here also gauge-readings 
are necessary, as the stage of the water in the lake varies from 
year to year. In this case the gauge need only be read twice 
a day. 

In rivers of variable stage the datum is either referred to 
mean or low-water stage, or else to the general datum of the 
map. If the stage is changing rapidly the gauge should be 
read hourly when soundings are taken, otherwise daily readings 
are sufficient. If the soundings are to be referred to the 
general datum of the map, then the slope of the stream must 
be taken into account. If they are referred to a particular 
stage of water in the river, then the slope does not enter as a 
correction, as the slope is assumed to be the same at all stages, 
although this is not strictly true. 

239. Lines of Equal Depth correspond to contour lines 
in topographical surveys ; but to draw lines of equal depth 
with certainty the elevations of many more points are neces- 
sary than are needed for drawing contour lines, because the 
bottom cannot be viewed directly, while the ground can be. 
Where the ground is seen to be nearly level no elevations 
need be taken, while for a similar region of bottom a great 
many soundings would be required to prove that it was not 
irregular. 

240. Soundings on Fixed Cross-sections in Rivers. — 
Where the same section is to be sounded a great many times, 
and especially when it is desirable to obtain the successive 
soundings at about the same points, it is best to fix range- 
posts on the line of the section (on both sides if it be a river) 



H YDROGRAPHIC S UR VE YING. 



289 



and then fix one or more series of intersecting ranges at points 
some distance above or below the section on one or both sides 
of the river. The soundings can then be made at the same 
points continuously without having to observe any angles at 
all. Such a system of ranges is shown in Fig. 72. AA f and 
BB' are range-poles on the section line. O and O' are tall 
white posts set at convenient points on opposite sides of the 
river, either above or below the section. I., II., III., etc., are 
shorter posts set near the bank in such positions that the in- 
tersection of the lines O-l., O-ll., etc., with the section range 




Fig. 72. 

BB' will locate the soundings at 1, 2, etc., on this section line. 
The posts in the banks should be marked by strips nailed upon 
them so as to make the Roman numerals as given in the figure. 
Such a system of ranges as the above is useful also for fixing 
points on a section-line, for setting out floats, or for running 
current-meters for the determination of river discharge. 

241. Soundings for the Study of Sand-waves. — In all 
cases where streams flow in sandy beds, the bottom consists 
of a series of wave-like elevations extending across the chan- 
nel. These are very gently sloping on the up-stream side 
19 



29O SUR VE YING. 



and quite abrupt on the lower side. They are called sand- 
waves, or sand-reefs. They are constantly moving down- 
stream from the slow removals from the upper side and accre- 
tions on the abrupt lower face. They have been observed as 
high as ten feet on the Mississippi River, and with a rate of 
motion as great as thirty feet per day. In order to study the 
size and motion of these sand-waves, it is necessary to take 
soundings very near together, on longitudinal lines over, the 
same paths at frequent intervals for a considerable period. 
The boat is allowed to drift with the current and the lead floats 
with the boat near the bottom. It is lowered to the bottom 
every few seconds and the depth and time recorded. About 
once a minute the boat is located by two instruments on shore 
or in the boat, and so the exact path of the boat located. A 
profile of the bottom can then be drawn for the path of the 
boat. A few days later the same line is sounded again in a 
similar manner and the two profiles compared. It will be 
found that the waves have all moved down-stream a short dis- 
tance, the principal waves still retaining their main charac- 
teristics, so that identification is certain.* 

242. Areas of Cross-section are obtained by plotting 
the soundings on cross-section paper, the horizontal scale be- 
ing about one tenth or one twentieth of the vertical. The 
horizontal line representing the water-surface is drawn, and the 
plotted soundings joined by a free-hand line. The enclosed 
area is then measured by the planimeter. If the horizontal 
scale is 50 feet to the inch and the vertical scale 5 feet to the 
inch, then each square inch of the figure represents 250 square 
feet of area. The planimeter should be set to read the area 
in square inches, and the result multiplied by 25o.f 

* It is believed the author made the first successful study of the size and 
rate of motion of sand-waves, at Helena, Ark., on the Mississippi River, in 
1879. See Rep. Chief of Engineers, U. S. A., 1879, vol. iii., p. 1963. 

f See p. 143 for a description and theory of the planimeter. 



HYDROGRAPHIC SURVEYING. 29 1 

Areas of cross-section are usually taken in running water, 
and here great care must be taken to get vertical soundings, 
and to make the proper sounding-line corrections. They 
should be taken near enough together to enable the bottom 
line to be drawn with sufficient accuracy. 

BENCH-MARKS, GAUGES, WATER-LEVELS, AND RIVER-SLOPE. 

•243. Bench-marks should be set in the immediate vi- 
cinity of each water-gauge, and these connected by duplicate 
lines of levels with the reference-plane of the survey. If the 
gauge is not very firmly set, or if it is necessary to move it for 
a changing stage, its zero must be referred again to its bench- 
mark by duplicate levels, whenever there is reason to suspect 
it may have been disturbed. Such bench-marks as these are 
usually spikes in the roots of trees or stumps. 

244. Water-gauges are of various designs, according to 
the situation and the purpose in view. For temporary use 
during the period of a survey, a staff gauge is best, consisting 
of a board painted white, of sufficient length, graduated to feet 
and tenths in black. Sometimes it is graduated to half-tenths, 
but this is useless unless in still water, and there is never any 
need of graduation finer than this. The gauge maybe read to 
hundredths of a foot if the water is calm enough. It should 
be nailed to a pile or to a stake driven firmly near the water's 
edge. It is read twice a day, or oftener, if the needs of the 
service require. 

For the continuous record of tidal stages an automatic, or 
self-registering, gauge is employed. For rivers with widely 
varying stage an inclined scantling is fixed to stakes set from 
low to high water along up the sloping bank. It should be 
placed at a point where the bank is neither caving away nor 
growing by filling-in of new deposits. After the scantling is 
set (the slopes not necessarily the same throughout its length), 
the foot and tenth graduations are set by means of a level and 



292 SUR VE YING. 



marked by driving copper tacks. The automatic gauge is 
described in Chap. XIV. The staff gauge is the one generally 
used for engineering and surveying purposes. 

245. Water-levels. — The surface of still water is by defi- 
nition a level surface. This fact is used to great advantage 
on the sea-coast, on lakes, ponds, and even on streams of little 
slope or on such as have a known slope. Thus, in finding the 
elevations of the Great Lakes above the sea-level, the elevation 
above mean tide- water of the zero of a certain water-gauge at 
Oswego, N. Y., on Lake Ontario, was determined. Then the rela- 
tive elevations of the zeros of certain gauges at Ports Dalhousie 
and Colborne, at the lower and upper ends of the Welland 
Canal respectively, were found by levelling between them, thus 
connecting Lake Ontario with Lake Erie. Lakes Erie and Hu- 
ron were joined in a similar manner by connecting a gauge at 
Rockwood, at the mouth of the Detroit River with one at Lake- 
port, at the lower end of Lake Huron. Lakes Michigan and 
Huron were assumed to be of the same level on account of 
the small flow between them and the very large sectional area 
of the Straits of Mackinac. Finally, a gauge at Escanaba, on 
Lake Michigan, was joined by a line of levels with one at Mar- 
quette, on Lake Superior. This completed the line of levels 
from New York to Lake Superior, when sufficient gauge-read- 
ings had been obtained to enable water- lev els \.o be carried from 
Oswego to Port Dalhousie, on Lake Ontario ; from Port Col- 
borne to Rockwood, on Lake Erie ; and from Lakeport, on 
Lake Huron, to Escanaba, on Lake Michigan. It was found 
that these water-levels were very accurate. Relative gauge- 
readings were compared for calm days, as well as for days 
when the wind was in various directions, and a final mean 
value found which in no case had a probable error as great as 

O.i foot.* 

• 

* See Primary Triangulation of the U. S. Lake Survey. 



HYDROGRAPHIC SURVEYING. 293 

A line of levels run along a lake shore or canal in calm 
weather should be checked at intervals by reading to the 
water-surface, and in a topographical survey the stadia-rod 
should frequently be held at the water-surface, even when the 
body of water is a stream with considerable slope, as it gives 
a check against large errors even then, and at the same time 
gives the slope of the stream. Mean sea-level at all points 
on the sea-coast is universally assumed to define one and the 
same level surface. It is probable, however, that this is not 
strictly true. Wherever a constant ocean current sets stead- 
ily against a certain coast, it would seem that the water here 
must be raised by an amount equal to the head necessary to 
generate the given last motion. If the current flows into an 
enclosed space, as the equatorial current into the Gulf of 
Mexico, or the tides into the Bay of Fundy, the water-surface 
may rise much higher. There is some evidence that the ele- 
vation of mean tide in the Gulf of Mexico is two or three feet 
higher than that of the Atlantic at Sandy Hook.* The evi- 
dence on this point is as yet insufficient to warrant any certain 
conclusion, however. 

246. River Slope is a very important part of a river survey. 
Sometimes it is desirable to determine it for a given stretch 
of river with great care, in which case it is well to set gauges 
at the points between which the slope is to be found and con- 
nect them by duplicate lines of accurate levelling. The gauges 
are then read simultaneously every five minutes for several 
hours and the comparison made between their mean readings. 
This is always done in connection with the measurement of the 
discharge of streams when the object is to find what function 
the discharge is of the slope. It is now known, however, that 
in natural channels the discharge is no assignable function of 

* See paper by Prof. Hilgard, Supt. U. S. C. and G. Survey, in Trans. Am. 
Asso. Adv. Sciences, 1884, p. 446. 



294 SUE VE YING. 



the slope, as is explained in section 259. For ordinary purposes 
the river slope may be determined with sufficient accuracy by 
simply reading the level or the stadia-rod at water-surface as 
the survey proceeds, daily readings of stage being made at 
permanent gauges at intervals of fifty miles or less along the 
river. 

In all natural channels the local slope is a very variable 
quantity. It is frequently negative for short distances in cer- 
tain stages, and over the same short stretch of river it may 
vary enormously at different stages, and even for the same 
stage at different times. It is determined by the local channel 
conditions, and these are constantly changing in streams flow- 
ing in friable beds and subject to material changes of stage. 
Great caution must therefore be exercised in introducing it 
into any hydraulic formulae for natural channels. It is usually 
expressed as a fraction, being really the natural sine of the 
angle of the surface to the horizon. That is, if the slope is one 
foot to the mile it is -g-gVo = 0.000189. 

THE DISCHARGE OF STREAMS. 

247. Measuring Mean Velocities of Water Currents. 

— This is usually done only for the purpose of obtaining the 
discharge of the stream or channel, but sometimes it is done 
for other purposes, as for the location of bridge piers or harbor 
improvements. In the case of bridge piers the direction of 
the current at different stages must be known, so that the piers 
may be set parallel to the direction of the current. For find- 
ing the discharge of the stream or other channel the object may 
be: 

(1) To obtain an approximate value of the discharge at the 
given time and place. 

(2) To obtain an exact value of the discharge at the given 
time and place. 

(3) To obtain a general formula from which to obtain sub- 






HYDROGRAPHIC SURVEYING. 295 

sequent discharges at the given place, or to test the truth of 
existing formulae, or to determine the relative efficiency of 
certain appliances or methods. 

It will be assumed that the second object is the one sought, 
and rnodified forms of the methods used to accomplish this 
may be chosen for other cases. 

The mean velocity of a stream is by definition the total dis- 
charge in cubic feet per second divided by the area of the 
cross-section in square feet. This gives the mean velocity in 
feet per second. Evidently this is the mean of the veloci- 
ties of all the small filaments (as of one square inch in area) on 
the entire cross-section. If the velocities of these filaments 
could be simultaneously and separately observed and their 
mean taken, this would be the mean velocity of the stream. It 
is quite impossible to do this ; but the nearer this is approached, 
the more accurate is the final result. If, however, we could 
obtain by a single observation the mean velocity of all the fila- 
ments in a vertical plane, the number of necessary observations 
would be diminished without diminishing the accuracy of the 
result. There are two common methods of measuring the ve- 
locities of filaments at any part of the cross-section, and one 
for obtaining at once the mean velocity in a vertical plane. 
These are by sub-surface floats and current-meters, and by rod 
floats, respectively. 

248. By Sub-surface Floats. — The ideal sub-surface float 
consists of a large intercepting area maintained at any depth 
in a vertical position by means of a fine cord joined to a sur- 
face float of minimum immersion and resistance, which bears 
a signal-flag. As good a form as any, perhaps, for the lower 
float, or intercepting plane, consists of two sheets of galvanized 
iron set at right angles, and intersecting in their centre lines, as 
shown in Fig. 73. There are cylindrical air-cavities along the 
upper edges and lead weights attached to the lower edges of 
the vanes. These serve to give the desired tension on the 



296 



SURVEYING. 



connecting cord and to maintain the float in an upright posi- 
tion, even though the cord is drawn out of the vertical by 
faster upper currents. The vanes should be from six to fifteen 
inches in breadth by from eight to twenty inches high, accord- 
ing to the size of the stream. The circular ribs serve simply 
to hold the vanes in place. The upper float is hollow, cylin- 




Fig. 73. 



drical in plan, and carries a small flag. The tension on the 
cord should be from one to five pounds, according to the size 
of the floats. The cord itself should be of woven silk and as 
small as possible, so as to exercise a minimum influence on the 
motion of the lower float. Wire is not suitable for this pur- 
pose, as it kinks badly in handling. The theory is that the 
lower float will move with the water which surrounds it, and 
that the upper float will be accelerated or retarded according 



HYDROGRAPHIC SURVEYING. 297 

as the surface current is slower or faster than that at the sub- 
merged float. The velocity of the current at any depth can 
thus be determined by running the lower float at this depth 
and observing the time required for the upper float to pass 
between two fixed range-lines at right angles to the direction 
of the current about two hundred feet apart. The floats are 
started about one hundred feet above the upper range-line, and 
picked up after having passed the lower range. Two transits 
are usually used for locating and timing the floats, one being 
set on each range. When the float approaches the upper 
range the observer on this line sets his telescope on range and 
calls " ready" as the float enters his field of view. The other 
observer then clamps his instrument and follows the float with 
the aid of the slow-motion or tangent screw. When the float 
crosses the vertical wire of the upper instrument he calls " tick," 
and the lower observer reads his horizontal angle. He then 
sets his telescope on the lower range while the upper observer 
follows the float with his telescope, and the operation is re- 
peated to obtain an intersection on the lower range. One or 
two timekeepers are needed to note the time of the two 
" tick" calls, the difference being the time occupied by the 
float in passing from the upper to the lower range-line. Both 
these signals are sometimes transmitted telegraphically to a 
single timekeeper. When the angles are plotted the path of 
the float is also obtained. 

If the channel is not too wide, wires may be stretched 
across the stream and the float stations marked on these, or 
the float stations may be determined by means of fixed ranges 
on shore. The passage of the floats across the section lines 
may then be noted by a single individual without a transit, 
using a stop-watch and possibly a field-glass. He starts the 
watch when the float reaches the upper section, walks to the 
lower section, and stops the watch when the float passes this 
range-line. The near range consists of a plumb-line, or wire 



298 



SURVEYING. 



suspended vertically; and the observer stands several feet back 
of this, and brings it in line with the range-post on the opposite 
side of the stream. 

If several floats are started a few minutes apart at the same 
station and at the same depth, they will sometimes vary as 




Fig. 74. 



much as twenty per cent in their times of passage, showing 
great irregularity in the velocity of different parts of the same 
filament. This is due to internal movements in the water, 
such as " boils," eddies, etc. It is for this reason that great 
refinement in such observations is useless. A float observation 



HYDROGRAPHIC SURVEYING. 



299 




1 1G. 75. 



gives only the velocity of a given small volume of water which 
surrounds the lower float, while a current-meter observation, 
as will be seen, gives the mean velocity of a given filament of 



300 SURVEYING. 



the stream of any required length. And as different portions 
of the same filament have very different longitudinal velocities, 
it requires a great many float observations to give as valuable 
information as may be obtained by running a current-meter 
in the same filament for one minute. 

If discharge observations are to be repeated many times at 
the same sections, then an auxiliary range should be established 
from which to start the floats; and if it is desirable to always 
run them over the same paths, these may be fixed by means 
of a system of intersecting ranges as described on p. 289. 

249. By Current-meter. — This is the most accurate method 
of obtaining sub-surface velocities ever yet devised. Three 
patterns of current-meters are shown in Figs. 74 and 75. 
The first and third are shown in elevation, together with the 
electrical recording-apparatus. The second is shown in plan. 
The first has helicoidal and the other two conical cup-shaped * 
vanes. Neither has any gearing under water, the record being 
kept by means of an electrical circuit which is made and bro- 
ken one or more times each revolution. The cup vanes are 
better adapted to water carrying fibrous materials which tend 
to collect on the moving parts. The friction can also be made 
less on the cup meters, agate or iridium bearings being used. 
The recording-apparatus is kept on shore or in a boat, while 
the meter is suspended by proper appliances at any point of 
the section at which the velocity of the current is to be measured. 
In deep water a boat, or catamaran, is anchored at the desired 



* Invented by Gen. Theo. G. Ellis, and first used on the survey of the 
Connecticut River. The telegraphic attachment is due to D. Farrand Henry 
of Detroit, Mich. See Report of the Chief of Engineers, U. S. A., 1878, p. 308. 

The form shown in Fig. 75 is due to W. G. Price, and was specially de- 
signed to be used on the Mississippi River. It is very strong and well pro- 
tected against floating drift. The first two forms are manufactured by Buff 
& Berger, of Boston, while the Price meter is made by W. & L. E. Gurley, 
Troy. 



HYDROGRAPHIC SURVEYING. 301 

point, and a weight attached to the meter, which is then lowered 
to the requisite depth by means of a windlass. After it is in 
place the connection is made with the battery, and the record 
kept for a given period of time, as for two or three minutes. 
If the operation is to be repeated often at the same section a 
wire anchorage laid across the stream above the line would be 
found useful. This wire is anchored at intervals and is used 
both for holding the boat (or catamaran) in place and for pull- 
ing it back and forth across the stream. In large rivers a 
steam-launch may be required for handling the catamaran.* 
In this case the record begins and ends when the observer is 
brought on range, it being impossible to hold up steadily 
against the current. If only the discharge of the stream is 
sought, the meter is run at mid-depth at a sufficient number 
of points in the section. 

The mean velocity in a vertical section at a given point may 
be obtained by moving the meter at a uniform rate from sur- 
face to bottom and back again, noting the reading of the regis- 
ter for the two surface positions, and also for the bottom posi- 
tion. If the boat was stationary and the rates of lowering and 
raising strictly constant and equal, the number of revolutions in 
descending and in ascending should be equal. Either of these 
registrations, divided by the time, would give the mean regis- 
tration per second of all the filaments in that vertical plane. 
The mean of the downward and upward results may be used 
as giving the mean velocity in that vertical plane. This will 
not be quite accurate, since it is impossible to run the meter 
very close to the bottom, but the results will be found useful 
for comparison with the mid-depth results. Such observations 
are sometimes called integrations in a vertical plane. 

250. Rating the Meter. — When any kind of current-meter 



* For a description of the latest methods used in gauging the Mississippi 
River see Report of the Miss. Riv. Com. for 1883, Appendix F. 



302 SURVEYING. 



is used for determining the velocity of passing fluids, only the 
number of revolutions of the wheel carrying the vanes is ob- 
served for a given time. Before the velocity of the fluid in feet 
per second can be found, the relation between the rate of revo- 
lution of the wheel and the rate of motion of the fluid must be 
determined for all velocities that are to be observed. The de- 
termination of this relation is called rating the meter. It is usu- 
ally done by causing the meter to move through still water at 




Fig. 76. 

a uniform speed, and noting the time occupied and the corre- 
sponding number of registrations made in passing over a given 
distance. It may be attached to the prow of a boat, as shown 
in Fig. 76, the electric register being in the boat. The dis- 
tance divided by the time gives the rate of motion or velocity 
of the meter through the water. The number of registrations 
(revolutions of the wheel) divided by the time gives the rate 
of motion of the wheel. The ratio of these two rates is the 
coefficient by which the registrations of the meter are trans- 
formed into the velocity of the current. This ratio is not a 
constant, but is usually a linear function of the velocity. Thus, 
if the observations be plotted, taking the number of registra- 
tions per second as abscissae and the velocities in feet per 
second as ordinates, they will be found to fall nearly in a right 
line, the equation of which is 

y =z ax -\- b . . . . (1) 



HYDROGRAPHIC SURVEYING. 303 

Here x and y are the observed quantities, while a and b are 
constants for the given instrument. If these constants could 
be found, then the values of y (velocity) could be obtained for 
all observed values of x (registrations). There are two ways of 
solving this problem — one graphical and one analytical. Evi- 
dently any two observations at different speeds would give 
values of a and b ; but to find the best or most probable values 
of these constants a great many observations are taken, so that 
we have many more observations than we have unknown quan- 
tities. Each pair of observations would give a different set of 
values of a and b. The most convenient method of finding 
the most probable values of these functions, though somewhat 
approximate, is 

(1) The Graphical Method of Solution. — This consists simply 
in plotting the corresponding values of x and y on coordi- 
nate paper, and drawing the most probable straight line through 
the points. Then the tangent of the angle this line makes 
with the axis of x is a, and the intercept on the axis of y is b. 
One point on this most probable line is the point (x y ), r Q and 
y being the mean values of the coordinates of all the plotted 
points. This is shown by equation (3). Having determined 
this point, a thread may be stretched through it and swung 
until it seems to be in a position of equilibrium, when each 
point is conceived as an attractive force acting on the line, the 
measure of the force being the vertical intercept between the 
point and the line. The arms of these forces are evidently 
their several abscissae. Or the forces may be measured by 
their horizontal intercepts, and then their arms are their seve- 
ral ordinates. For the position of equilibrium the sum of the 
moments of these forces about the point (x y ) would be zero. 
Such a determination of a and b would be found sufficiently 
accurate for all practical purposes, but if desired the problem 
may be solved by 



304 SURVEYING. 



(2) The Rigid or Analytical Method. — Equation (2) may be 
written 

b -f- xa — y = o. 

Every observation may be written in this form, these being 
called the observation equations. It is probable that no given 
values of a and b would satisfy more than two of these obser- 
vations ; and if the most probable values be used, there would, 
in general, be no single equation exactly satisfied. If we -let 
x v x„ Gtc. t y v y v etc., and v v v„ etc., be the several values of 
.r, y, and the corresponding residuals for the several observa- 
tion equations, we would have 



b + x x a—y 1 — v 1 \ 1 

b + x i a-y i = v i ; ! ^ 

b + x m a—y n = v m .j 



Since b enters alike in all of them, it is evident that these 
equations are all of equal value for determining b. Also, since 
the properly weighted arithmetic mean is the most probable 
value of a numerously observed quantity, and since in this case 
the equations (or observations) have equal weight for deter- 
mining b, we may form from the given series of equations a 
single standard or " normal " equation which will be the arith- 
metic mean of the observation equations ; put this equal to zero 
and say this shall give the value of b. If x a and y be the mean 
values of the observed x's and ys, we would then have, by add- 
ing the equations all together and dividing by their number 

b + x a— y = o, or b=y — x a. ... (3) 



HYDROGRAPHIC SURVEYING. 305 

Substituting this value of b in equation (2), we have 

(jr, - x )a - (y x -y ) =v,; } 
(^-x )a-(y 2 -y ) = v 2 ; I .... (4) 

{*m — *o)a — {fm —fo) = V m . J 

We here have a series of equations involving one unknown 
quantity ; but they evidently are not of equal value in deter- 
mining the unknown quantity a, since its coefficients are very 
different. In fact, the relative value of these equations for de- 
termining a is in direct proportion to the size of this coefficient, 
so that if this coefficient is twice as large in one equation as in 
another, the former equation has twice the value of the latter 
for determining a. In other words, they should all be weighted 
in proportion to the values of these coefficients, and a conve- 
nient way of doing this is to multiply each equation through 
entire by this coefficient. The resulting multiplied equations 
then have equal weight, and may then be added together to 
produce another " normal" equation for finding a. This result- 
ing equation is 

[(* - *.)"> - [(* - *„ (y -a)] = o, . . . (5) 

where [ ] is a sign of summation. If we had divided this 
equation by the number of observation equations m, it would 
in no sense have changed it so far as the value of a is concerned. 
From equation (5) we can find the mean or most probable 
value of a, which when substituted in (3) gives the most prob- 
able value of b. These values should agree very closely with 
those found by the graphical method. The analytical method 
here given is precisely that by least squares, though arrived at 
through the conception of a properly weighted arithmetic mean, 
instead of by making the sum of the squares of the residuals a 
minimum. 
20 



306 



SURVEYING. 



The following is an actual example from the records of the 
Mississippi River Survey: 

REDUCTION OF OBSERVATIONS FOR RATING METER A, 

taken at Paducah, Ky., June 21, 1882. 

W. G. Price, Observer. L. L. Wheeler, Computer.* 



No. 


r 
IOO 


t 
53 


X 


y 


X — X Q 


y—y\ 


(-r-.ro) 2 


{x — x ) 

iy-yo) 


Remarks. 


I 


1 886 


3-774 


4- 0.117 


+ 0.245 


-f- 0.014 


-f- 0.029 


Observations 


2 


IOI 


44 


2.295 


4.544 


4- 9.526 


+ x .oi5 


+ 0.277 


+ 0.534 


made with 


3 


IOI 


4i 


2.464 


4.878 


4- 0.695 


+ 1-349 


4- 0.483 


+ 0.938 


meter on vertical 


4 


96 


124 


774 


1. 613 


- 0.995 


— 1. 916 


-\- 0.990 


+ 1.906 


iron rod, five 


5 


94 


152 


0.618 


1. 316 


- i.151 


— 2.213 


4- 1.325 


-f 2.548 


feet in front of 


6 


90 


193 


0.466 


1.036 


- 1.303 


- 2.493 


-f- 1.697 


+ 3-249 


bow of skiff, in 


7 


9i 


181 


0.503 


1. 105 


— 1.266 


— 2.424 


+ 1.603 


+ 3-069 


pond. 


8 


103 


28 


3.678 


7.142 


+ 1.909 


+ 3.613 


4- 3.644 


+ 6.903 




9 


IOO 


53 


1.886 


3-774 


-f- 0.117 


4" 0.245 


+ 0.014 


-J- 0.029^ 


Length of base 


10 


98 


73 


T.342 


2.740 


- 0.427 


- 0.789 


-1- 0.182 


+ o.337 


= 200 feet. 


11 


103 


29 


3-552 


6.896 


4- 1.783 


+ 3-367 


+ 3.178 


4- 6.002 




[x 


] = 19.464 


38.818 =[>] 


[(*-.*o) 2 ]= 13.407 


25 -544= [(*— ^0X^—^0)] 




X 


— 


1.769 


3.529 


=yo 











Normal Equations, 
b -f- 1.769a — 3529 = o ; Whence a = 1.905 ; 

13 • 4070 - 25-544 = o. £ = 0.159. 

Equation for Rating, 
y = 1.905*4-0.159. 

Even where the analytical method is to be used it is al- 
ways well to plot the observations for purposes of study. 
Then if any observations are especially discrepant, the fact will 
appear. By consulting column six of the computation it will 



* In the original computation the method by least squares was used and the 
probable errors of a and b found. 



HYDROGRAPHIC SURVEYING. 307 



be seen that observations of greatest weight were those taken at 
very high and at very low velocities. If the observations were 
taken in three groups about equally spaced, an equal number 
of observations in each group, the members of a group being 
near together, then the mean of each group could be used as 
a single observation. The middle group would serve to show 
whether or not the unknown quantities were linear functions of 
each other, since, if they were, the three mean observations 
should plot in a straight line. The value of a could be com- 
puted from the two extreme mean observations, and the value 
of b from the mean of all the observations as before. This 
would give a result quite as accurate as to treat them separately, 

If the observations do not plot in a straight line, draw 
the most probable line through them, and prepare a table of 
corresponding values of x and y from this curve. In any case, 
a reduction table should be used. 

The meter should be rated frequently if accurate results are 
required. In the rating the meter should be fastened several 
feet in front of the bow of the boat, and in its use it should be 
run at a sufficient distance from the boat or catamaran to be 
free from any disturbing influence on the current. 

251. By Rod Floats. — These may be either wooden or tin 
rods, of uniform size, loaded at the bottom, and arranged for 
splicing if they are to be used in deep water. If the channel 
were of uniform depth, and the rod reached to the bottom with- 
out actually touching, then the velocity of the rod would be 
the mean velocity of all the filaments in that vertical plane,* 



* This is not strictly true, since the pressure of a fluid upon a body moving 
through it varies as the square of its relative velocity. The rod moves faster 
than the bottom filaments and slower than the upper filaments, but this differ- 
ence Is greatest at the bottom. Therefore, the retarding action of the bottom 
filaments will have undue weight, as it were, and so the velocity of the rod will 
really be about one per cent slower than the mean velocity of the current. See 
"Lowell Hydraulic Experiments," by James B. Francis. 



308 SUR VE YING. 






and this is the value sought. In practice the rod can never 
reach the bottom, even in smooth, artificial channels, while in 
natural channels the irregularities are usually such as prohibit 
its use within several feet of the bottom. The methods of 
observation are the same as with the double floats, and their 
velocity is the mean velocity of the water in that plane to the 
depth of immersion. For artificial channels, and for natural 
channels not more than twenty or thirty feet deep, rod floats 
may be advantageously used. Beyond that depth they cannot 
be made of sufficient length to give reliable results. The 
method is, therefore, best adapted to artificial channels of uni- 
form cross-section. 

The immersion of the rod should be at least nine tenths of 
the depth of the water, in which case, and for uniform channels, 
as wooden flumes, Francis found that the velocity of the rod 
required the following correction to give the mean velocity of 
the water in that vertical plane : 

V m •= V r [I—0.II6(^-0.I)]. 

Where V m = mean velocity in vertical plane ; 
V r = observed velocity of rod ; 

^ depth of water below bottom of rod 

depth of water 

For natural channels, or for a less immersion than nine- 
tenths of the depth the formula cannot be used with certainty. 
The rods should be put into the water at least twenty feet 
above the upper section. 

252. Comparison of Methods. — (1) The method by double 
floats is adapted to large and deep rivers, or rapid currents 
carrying much drift or impeded by traffic. It may be used 
in all cases, but it has the disadvantage of registering only the 
velocity of a small volume of water surrounding the lower 
float. 



HYDROGRAPHIC SURVEYING. 309 

(2) The method by meters is adapted to large or small 
streams. It records the mean velocity of a filament of indefinite 
length ; but it cannot be used where the water carries consider- 
able floating debris, or where the current is too swift to admit 
of a safe anchorage. 

(3) The method by rods is best adapted to small channels 
of uniform section ; it records the mean velocity in a vertical 
plane to a depth equal to its immersion, and it can be univer- 
sally used when the law of the velocities in a vertical plane is 
known, for then a proper coefficient could be derived for any 
depth of immersion. 

(4) One rod observation of sufficient immersion is prob- 
ably as good as several float observations, and a current-meter 
observation of two or three minutes is worth as much as 
twenty float observations for the same filament, provided the 
meters rate is constant and well determined. 

(5) The rods and floats are cheaper in first cost than the 
meter ; but if the work is to be prosecuted for a considerable 
period, the excess in the cost of the outfit will be more than 
balanced by the diminished cost of the work, by using the 
meter. On the whole, it may be said that the method by cur- 
rent-meter is the most accurate and satisfactory of any yet de- 
vised for measuring the velocity of running water. 

523. The Relative Rates of Flow in Different Parts of 
the Cross-section. — (i) In a horizontal plane. If the cross- 
section of a stream were approximately the segment of a circle, 
then the relative rates of flow of the different filaments in any 
horizontal plane would be very nearly represented by the ordi- 
nates to a parabola, the axis of the parabola coinciding with 
the middle of the stream. If there should be any shoaling in 
any part of this ideal section the corresponding ordinates would 
be shortened, so that when the curve of the bottom is given 
the curve of velocities in a horizontal plane can be fairly pre- 
dicted. This applies only to straight reaches. If a portion of 
the section has a flat bottom line, the velocities over this por- 



3io 



SURVEYING. 



tion will be about uniform. Where the depth is changing rap- 
idly on the section, there the velocities will be found to change 
rapidly for given changes in positions across the section. 

It follows from this that observation stations should be 
placed near together where the section has a sloping bottom 
line, and they may be placed farther apart where the bottom 
line of the section is nearly flat. They are usually put closer 
together near the bank than near the middle of the stream. 

(2) In a vertical plane. A great deal of time and talent has 
been spent in trying to find the law of the relative rates of flow 



















































1 1 1 


1 1 












































-- 


__ 


-_ 


— 


(a 


lv 


T in 


dc 


01 


-w_ 


\~ 


.-- 






















































































































































































































































H 


'al_ 








\ 










































- 


— 


-- 


- 




\S- 




-- 


i 


1 
































































































































































































' 




















































(-V 


jw 


in 


Lt 


P 


























































7 
































































/ 
































































































































/ 






























































1 
































































1 


, 




























































Cl l 


c ! 


u 




























































































































1 


1 







0' 



3' 



5' 



Fig. 77. 



6' 



in a vertical plane, but there is probably no law of universal 
application. The curve representing such rates of flow will 
always resemble a parabola more or less, the axis of which is 
always beneath the surface except when the wind is down 
stream at a rate equal to, or greater than, the rate of the cur- 
rent. That is to say, the maximum velocity is always below 
the surface except where the surface filaments are accelerated 
by a down-stream wind, and it is generally found at about one 
third the depth. The cause of this depression of the filament 
c f maximum velocity is partly due to the friction of the air, 



HYDROGRAPHIC SUR VE YING. 



3" 



but mostly to an inward surface flow from the sides toward 
the centre, which brings particles having a slower motion 
towards the middle of the surface of the stream. This inward 
surface flow is probably due to an upward flow at the sides 
caused by the irregularities of the bank, which force the parti- 
cles of water impinging upon them in the direction of the least 
resistance which is vertical.* The curves in Fig. 77 represent 
the mean vertical curves of velocity observed at Columbus, 
Ky., on the Mississippi River and given in Humphreys and 



Scale &f feet. 



OEKXE 




Fig. 78. 

Abbot's Report. The left-hand vertical line is the axis of ref- 
erence, and the curves are found to fall between the seven- and 
eight-foot lines. That is, the velocity at all depths in this 
plane was between seven and eight feet' per second. In this 
case double floats were used, and it is probable that the bottom 
velocities were not very accurately obtained. The effect of the 
wind is here shown in shifting; the axis of the curve. It is to 



See paper by F. P. Stearns before the Am. Soc. Civ. Engrs., vol. xii. p. 331. 



3 1 2 SUR VE YING. 



be observed that these curves all intersect at about mid-depth. 
That is to say, the velocity of the mid-depth filament is not 
affected by wind. This is why the mid-depth velocity should 
be chosen when the velocity of but a single filament is to be 
measured, and from this the mean velocity in the vertical sec- 
tion derived. It has also been found that the mid-depth veloc- 
ity is very near the mean velocity, being from one to six per 
cent greater, according to depth and smoothness of channel. 
In general, for channels whose widths are large as compared 
to their depths, a coefficient of from .96 to .98 will reduce 
mid-depth velocity to the mean velocity in that vertical plane. 

In Fig. J%* are shown the relative velocities in different parts 
of the Sudbury River Conduit of Boston. The velocity at 
each dot was actually measured by the current-meter. The 
lines drawn are lines of equal velocity, being analogous to con- 
tour lines on a surface, the vertical ordinates to which would 
represent velocities. The method of obtaining these velocities 
is shown in Fig. 79. B is a pivoted sleeve through which the 
meter-rod slides freely. At A there is a roller fixed to the rod 
which runs on the curved tracks a a a. The graduations on 
these tracks fix the different positions of the meter, these be- 
ing so spaced that they control equal areas of the cross-section. 
Integrations were here taken in horizontal planes by moving the 
meter at a uniform rate horizontally. 

254. To find the Mean Velocity on the Cross-section. 
— It is evident that this mean velocity cannot be directly ob- 
served. In fact, it can only be found by first finding the dis- 
charge per second and then dividing this by the total area of 
the section. That is to say, the mean velocity is, by definition, 



Q 



v = 2 . 



* This and the following figure are taken from the paper by F. P. Stearns, 
mentioned in foot-note on the previous page. 



HYDROGRAPHIC SURVEYING. 



313 



umXREXTT 




Fig. 79. 



3 H SURVEYING. 



The area of the section is found by means of properly located 
soundings. The actual velocities of certain filaments crossing 
this section are then observed, and the section subdivided in 
such a way that the observed velocities will fairly represent 
the mean velocities of all the similar filaments (usually mid- 
depth) in that subsection. Each observed velocity is then 
reduced to the mean velocity in that vertical plane, and this is 
assumed to be the mean velocity in that subsection. These 
mean velocities, multiplied by the areas of their corresponding 
sections, give the discharges across these sections, and the sum 
of these partial discharges is the total discharge, Q, in the 
above equation. This may be shown algebraically as follows : 

Let V v V„ V a , etc., be the observed velocities ; 

C the coefficient to reduce these to the mean velocity 
in a vertical plane ; 

A x , A„ A z , etc., the partial areas of the cross-section 
corresponding to the observed velocities V x , V„ 
F 3 , etc. ; 

A the total area of the cross-section = A x -f- A 2 -f- A z1 
etc.; 

Qv Q» Q*> etc v the partial discharges ; 

Q the total discharge ; 

v the mean velocity for the entire section. 

Then Q x = CV X A X \ Q* = CV,A„ etc. ; 

G = & + & + etc. = C{Ay x + A,V, + etc.); 



and v = f = *%A X V x + A 7 K + etc.). 



HYDROGRAPHIC SUR VE YING. 



315 




Fig. 8a. 



316 SURVEYING. 



It has been here assumed that observations are made at but 
one point in any vertical plane. The method is the same, how- 
ever, in any case, it only being necessary to apply such a co- 
efficient to the observed velocity as will reduce it to the mean 
velocity in its own sub-area. If these partial areas are made 
small, as in the case of the Boston Conduit, the observed ve- 
locities may be taken as the mean velocities in those areas ; and 
if these areas are all equal, which was also the case in this con- 
duit, then the mean velocity is the arithmetic mean of all the 
observed velocities. The partial and total areas are best found 
by means of the planimeter, the cross-section having been 
carefully plotted on coordinate paper. 

255. Sub-currents. — It is often desirable to know the 
direction as well as the velocity of flow beneath the surface. 
The direction-meter,* Fig. 80, is designed to give the direction 
of sub-currents, both horizontally and vertically. A magnetic 
needle swings freely until lifted and held by a magnet which 
is operated from above. The vertical direction is recorded in 
a similar manner on a small index-circle on the inside of the 
hollow sphere which always maintains an upright position. 

256. The Flow of Water over Weirs.f — The most ac- 
curate mode of measuring the flow through small open channels 
is by means of weirs. There are three kinds of weirs with 
which the engineer may have to deal in measuring the flow of 
water, — namely, sharp-crested weirs, wide-crested weirs, and 
submerged weirs. 

A sharp-crested weir is one which is entirely cleared by the 
water in passing over it, as in Fig. 81. A wide crest is shown 

* Invented by W. G. Price and manufactured by Gurley, Troy, N. Y. 

\ The results given in this and the following article have been mostly taken 
from a paper by Feeley and Stearns before the Am. Soc. Civ. Engrs., vol. xii. 
(1883), describing experiments made in connection with the new Sudbury River 
Conduit, Boston, Mass. The paper was awarded the Norman medal of that 
society. 



HYDROGRAPHIC SUR VE YING. 



317 





3i8 



SURVEYING. 



in Fig. 82, and it seffect in increasing the depth on the weir 
for a given discharge. If the crest has a width equal to the 
line ab in Fig. 81, then the depth on the weir is unaffected, 
while if it has a less width, as in Fig. 83, and if the air has not 
free access to the intervening space beneath, the water will soon 
fill this space, and the tendency to vacuum here will depress 
the overflowing sheet of water, thus diminishing the depth on 
the weir for a given flow. The dotted lines in Fig. 84 are 




Fig. 85. 

those of normal flow, the full lines being the new positions 
assumed as a result of the partial vacuum below. 

A submerged weir is one at which the level of the water 
below the weir is above its crest, there being, however, a certain 
definite fall in passing the weir, as shown in Fig. 85. Here 
h = d — d ' is the fall in passing the weir. 

Velocity of Approach. — This is the velocity of the surface- 
water towards the weir at a distance above the weir equal 
to about two and one half times the height of the weir above 
the bottom of the channel. 

End Contractions. — These are the narrowing effects of the 
lateral flow at the ends of the weir. If this lateral component 
of the flow is shut off by a plank extending several feet up 
stream and from the water's surface to several inches below 
the top of the weir, then there is no end contraction. This 
arrangement gives more accurate results, as the correction for 
end contraction involves some uncertainties. 



HYDROGRAPHIC SURVEYING. 



319 



Depth of Water on the Weir. — This is the principal function 
of the discharge ; it is the difference of eleva- 
tion between the top of the weir and the surface 
of the water at a distance above the weir equal to 
about 2^ times the height of the weir above the 
bottom of the channel. Evidently this is a quan- 
tity which cannot be directly measured. The 
best way of measuring this quantity is as follows : 
At a convenient point arrange a closed vertical 
box which connects by a free opening with the 
channel at about mid-depth at a point some six 
feet above the weir. The water will then stand 
in this box at its normal elevation, unaffected by 
the slope towards the weir. The elevation of 
this water-surface is determined by means of a 
hook-gauge, Fig. 86, which consists of a metallic 
point turned upwards and made adjustable in 
height by means of a thumb-screw. When the 
point of the hook comes to the surface of the 
water it causes a distorted reflection. The eleva- 
tion of the water-surface can be found in this 
way with extreme accuracy. The difference of V 

elevation between the point of the hook and the W 

crest of the weir can then be determined with a F ^. 86. 
level and rod. This difference is If in the following formulae. 

257. Formulae and Corrections. — For a simple sharp- 
crested weir, without end contractions and with no velocity of 
approach, the discharge in cubic feet per second is 



Q = 3.31LHS -f- 0.007Z, 



(0 



where L is the length of the weir and H the depth of water 
upon it, both measured in feet. The weir must have a level 
crest and vertical ends ; it should be in a dam vertical on its 



320 SURVEYING. 



up-stream side ; the water on the down-stream side may stand 
even with the crest of the weir if it has considerable depth. 
The error is not more than one per cent when the water on the 
down-stream side covers fifteen per cent of the weir area, pro- 
vided H is then taken as the difference in elevation of the 
water-surface above and below the weir. In this case two- 
hook gauges would be needed. The crest of the weir should 
be at a height above the bottom of the channel on the up- 
stream side equal to at least twice the depth on the weir, to 
allow for complete vertical contraction. 

The following corrections apply to their respective condi- 
tions : 

For the velocity of approach, the depth on the weir, H in 
equation (i), is to be increased by 1.5^, where there is no end 

contraction, h being the head due to the velocity, or h = — . 

2 S 



At sea-level this correction becomes 



.2 



C - — - = 0.0234^. (2) 



This is to be added to H in equation (1), v being measured in 
feet per second. 

Where there is end contraction, the correction is 



_ 2.05^ . . . 

C = -— = 0.032*/' (3) 

z <5 



For end contraction, the length of the weir, L in equation 
(1), is to be shortened by 0.1H for each such contraction. This 
is a mean value, although it varies from o.ojH to 0.12H for 
different depths on the weir varying from 1 to 0.3 foot, the 
smaller correction applying to the greater depth on the weir. 



HYDROGRAPHIC SURVEYING. 321 

For wide crests the correction to the depth on the weir is 
sometimes positive and sometimes negative, as shown in fig- 
ures 82 and 84. The following correction is derived from care- 
ful experiments : 



c = 0.2016 vy -f- 0.2 146^ — 0.1 876 w, . . . (4) 

where 

C is the correction to be added algebraically to the depth 
on the wide crest to obtain the depth on a sharp crest 
which will pass an equal volume of water ; 

w is the width of the crest ; 

y is the difference between o.Soyw and the depth on the 
crest. 

If the crest is narrower than the line ab, Fig. 81, then this 
correction is not to be applied unless the water adheres to the 
weir as. in Fig. 84. 

Up-stream edge of the weir rounded. If the up-stream edge 
of the weir is a small quarter-circle, add seven tenths of its ra- 
dius to the depth on the weir before applying the general weir 
formula. 

Submerged weir. When the water on the down-stream 
side rises above the level of the crest, use the formula for a 
submerged weir, which is 

Q = cl(<t+f)VI, (5) 

where 

Q is the discharge in cubic feet per second ; 

c is to be taken from the following table, its value varying 

with -7 ; 
d 

I is the length of the weir in feet ; 
21 



322 



SURVEYING. 



d is the depth on the weir in feet, measured from still 

water on the up-stream side ; 
d! is the depth to which the weir is submerged, measured 

from still water on the down-stream side ; 
h is the fall and equals d — d '. 

The value of d may be corrected for velocity of approach 
by formulas (2) and (3). There is no known correction for the 
velocity of discharge below the weir, and hence the formula 
can only be used for a channel of large capacity below, as com- 
pared with the discharge, so that the velocity here will be small. 

The following are the experimental values of £\ 



d' 
d' 


c. 


d' 
d' 


c. 


d' 
d' 


c. 


df 

d' 


c. 


O.OI 


3-330 


0.25 


3-249 


0.55 


3.100 


O.85 


3-I50 


.05 


3-36o 


.30 


3.214 


.60 


3-092 


.90 


3.I9O 


.08 


3-372 


-35 


3.182 


-65 


3.089 


•95 


3.247 


.IO 


3-305 


.40 


3-155 


.70 


3.092 


1. 00 


3-360 


•15 


3-327 


•45 


3-I3I 


•75 


3.102 






.20 


3.286 


•50 


3. 113 


.80 


3.122 







d' 

This table is inapplicable to values of -j less than 0.08, un- 
less the air has free access to the space underneath the sheet. 

The method of measuring discharge by means of sub- 
merged weirs is adapted to channels having very small slope. 
A fall as low as one half inch will give reliable results if it is 
accurately measured. 

258. The Miner's Inch. — This is an arbitrary standard 
both as to method and as to volume of water discharged. It 
rests on the false assumption that the volume discharged is 
proportional to the area of the orifice under a constant head 
above the top of the orifice. Its use grew out of the necessities 



HYDROGRAPHIC SURVEYING. 3^3 

of frontier life in the mining regions of the West, and should 
now be discarded in favor of absolute units. The miner's inch is 
the quantity of water that will flow through an orifice one inch 
square, under a head of from four to twelve inches, according to 
geographical locality. Even if the head above the top of the 
orifice be fixed, and a flow of 144 miner's inches be required, 
the volume obtained would be 3.3, 4.2, or 4.7 cubic feet per 
second, according as there were 144 holes each one inch square, 
one opening one inch deep and 144 inches long, or one opening 
twelve inches square, the tops of all the openings being five 
inches below the surface of the water. This simply illustrates 
the unreliable nature of such a unit. In some localities the 
following standard has been adopted : An aperture twelve 
inches high by twelve and three-quarter inches wide through 
one one- and one-half-inch plank, with top of opening six inches 
below the water-surface, is said to discharge two hundred 
miner's inches. By this standard the miner's inch is 1.5 cubic 
feet per minute, or 2160 cubic feet in twenty-four hours. 
Other standards vary from 1.39 to 1.78 cubic feet per minute.* 
When the miner's inch can only be defined as a certain num- 
ber of cubic feet per minute, it is evidently no longer of ser- 
vice and should be abandoned. The method by weirs is more 
accurate, and could almost always be substituted for the 
method by orifices. 

259. The Flow of Water in Open Channels. — For more 
than a century hydraulic engineers have labored to find a fixed 
relation between the slope and cross-section of a running stream 
and the resulting mean velocity. If such a relation could be 
found, then the discharge of any stream could be obtained at N 
a minimum cost. It is now known that there is no such fixed 
relation. There certainly is a relation between the bed of a 
stream for a considerable distance above and below the section, 

* See Bowie's " Hydraulic Mining," p. 126 (John Wiley & Sons, New York). 



324 SURVEYING. 



the surface slope, and the resulting velocity at the section ; 
but as no two streams have similar beds, nor the same stream 
in different portions of its length, and since the bed character- 
istics are difficult to determine, and, furthermore, are constantly- 
changing in channels in earth, the function of bed cannot be 
incorporated into a formula to any advantage except for chan- 
nels of strictly uniform and constant bed, in which case the 
cross-section would sufficiently indicate the bed. Again, the 
slope cannot be profitably introduced into a velocity formula 
except where it is uniform for a considerable distance above 
and below the section, for the inertia of the water tends to 
produce uniform motion under varying slopes, and the effect 
is that the velocity at no point corresponds strictly to the 
slope across that section. For uniform bed and slope, how- 
ever, formulae may be often used to advantage. 

Let A = area of cross-section ; 

v = velocity in feet per second (— / for one second) ; 
p S3s wetted perimeter ; 

A 

r = hydraulic mean radius = — ; 

P 

Z 

s = surface-slope = sin z = — ; 

Z = fall per length /; 

Q = quantity discharged in one second ; 

S == wetted surface in length / = //; 

f = coefficient of friction per unit area of S; 

p = weight of one cubic foot of water === density. 

Since the friction varies directly as the density and as the 
square of the velocity, we have for the frictional resistance on 
the mass covering the area S, 

R=fpS*, (i) 



HYDROGRAPHIC SURVEYING. 325 

and the work spent in overcoming this resistance in one sec- 
ond of time is 

K= Rv = fpSv* (2) 

If the velocity is constant, which it is assumed to be, then 
this is also the measure of the work gravity does on this mass 
of water in pulling it through the height h! — h" == Z, which 
work is 

K = weight X fall = ZpQ — ZpvA ; . . . (3) 
.-. ZpvA = fpSv\ (4) 

*»<§* ■ - (5) 



A Z 

But S •= pl\ — == r\ and —, = sin i === &; 

P l 



.*. s ±= - — or v — € Vrs, .... (6) 



where c is an empirical coefficient to be determined. It is evi- 
dent that c is mostly a function of the character of the bed, 
and that it can, therefore, have no fixed value for all cases. 

Equation (6) is what is known as the Chezy formula. The 
most successful attempt yet made to give to the coefficient c 
a value suitable to all cases of constant flow is that of Kut- 
ter.* Kutter's formula, when reduced to English foot-units, is 

* Kutter's Hydraulic Tables, translated from the German by Jackson, and 
published by Spon, London, 1876, 



326 



SUR VE YING. 



v = c vrs 



Vrs = 



^ , 1.811 , 0.00281 

4I-6+ — — + 



n 



1 + U1.6+ 



0.0028 1 \ n 



Vr 



Vrs, 



■ (7) 



the total coefficient of the radical, in brackets, being the eval- 
uation of c in equation (6). Here v, r, and s are the same 
as before, and n is a " natural coefficient " dependent on the 
nature of the soil, character of bed, banks, etc. Although it 
was the author's intention to make a formula that would be 
applicable even to natural channels, it cannot safely be ap- 
plied to such unless they have great uniformity of bed and 
slope. 

The following values of n are given by Kutter: 



Planed plank, 




n — 0.008. 


Pure cement, 




n = .009. 


Sand and cement, 




n = .010 to .011. 


Brickwork and ashlar, 




n = .012 " .014. 


Canvas lining, 




n = .015. 


Average rubble, 




n = .017. 


Rammed gravel, 




n == .020. 


In earth — canals and ditches, 


n — .020 to .030, 






depending on the reg- 






ularity of the cross- 






section, freedom from 






weeds, etc. 


In earth of irregular cross- 


■section, 


n == .030 to .040. 


For torrential streams, 




n = .050. 



In the last two cases the results are very uncertain. Kut- 
ter' s tables are evaluated for n = 0.025, .030, and .035. 



HYDROGRAPHIC SURVEYING. Z 2 7 

The greatest objection to the use of this formula is the 
labor involved in evaluating the " c" coefficient. To facilitate 
the use of the formula this coefficient has been evaluated for a 
slope of o.ooi in Table * VIII. This coefficient changes so 
slowly with a change in slope that the error does not exceed 
3J per cent if the table be used for all slopes from one in ten 
to one in 5280, which is a foot in a mile. These tabular co- 
efficients may therefore be used in all cases of ditches, pipe- 
lines, sewers, etc. The coefficients are seen to change rapidly 
for different values of n, so this value must be chosen with 
care. 

For brick conduits, such as are used for water-supply and 
for sewers, the formula 

v = i22r°' 62 s°- 5 

was found to represent the experiments on the Boston con- 
duit, shown in Figs. 78 and 79. This would correspond to a 
variable value of n in Kutter's formula, being nearly 0.012 
however, as given for brickwork. This conduit is brick-lined. 
Table IX.* gives maximum discharges of such conduits 
as computed by Kutter's formula, n being taken as 0.013. 
The results in heavy type include the working part of the 
table for sewers. All less than three feet per second when the 
depth of water is one eighth of the diameter, or when the flow 
is one fiftieth the maximum. This is as small a velocity as is 
consistent with a self-cleansing flow in sewers. All values 
below the heavy-faced type correspond to velocities more than 
fifteen feet per second when the conduit runs full, and this is 
as great a velocity as is consistent with safety to the structure. 
If the velocity is greater than this, the conduit should be lined 
with stone. « 

* Taken from a paper by Robt. Moore and Julius Baier in Journal of the 
Association of Engineering Societies, vol. v., p. 349. This table may also be 
used for tile drains. 



328 



SURVEYING. 



The maximum flow does not occur when the conduit runs 
full, but when the depth is about 93 per cent of the diameter. 
A conduit or pipe will therefore not run full except under 
considerable pressure or head. The maximum velocity occurs 
when the depth is about 81 per cent of the diameter. 

The relative mean velocities and discharges of a circular 
conduit for varying depths is shown by the following table: 



Depth of 
Water. 


Relative 
Velocity. 


Relative 
Discharge. 


Depth of 
Water. 


Relative 
Velocity. 


Relative 
Discharge. 


.1 


.28 


.016 


•7 


.98 


.776 


.2 


.48 


.072 


•75 


•99 


.850 


•25 


•57 


.118 


.8 


•99 


.912 


•3 


.64 


.168 


.81 


1. 00 


.924 


•4 


.76 


.302 


•9 


.98 


.992 


•5 


.86 


.450 


•93 


.96 


I. OOO 


.6 


•93 


.620 


1. 00 


.86 


.916 



260. Cross-sections of Least Resistance. — From equa- 
tion (6) of the preceding article it is apparent that for a given 
channel the velocity varies as the square-root of the hydraulic 

A 
mean radius, r. But r = — , hence for a given area of cross- 

P 
section the velocity is greater as the wetted perimeter is less. 

The form of cross-section having a minimum perimeter for a 
given area is the circular, or for an open channel the semicircu- 
lar. In both cases the hydraulic mean radius is r = ■ - = — , 

27rR 2 

where R is the radius of the circle. Since it is not always con- 
venient to make the cross-section circular in the case of ditches 
and canals, it is evident that the more nearly a polygonal 
cross-section coincides with the circular form the less will be 
the resistance to flow. When a maximum flow is desired for 




HYDROGRAPHIC SURVEYING. 329 

a given slope and cross-section, therefore, the shape should 
conform as nearly as possible to that of a semicircle. To do 
this, construct a semicircle to scale of the required area of 
cross-section. Draw tangents for the 
sides of the section having the de- 
sired slope and join these by another 
tangent line at bottom, as in Fig. 
87. This gives a little larger section- 
al area, but some allowance should Fig. 87. 
be made for accumulations in the 

channel. If the slope is very great and it is desirable to re- 
duce the velocity of flow, it may be done by making the 
channel wide and shallow. 

261. Sediment-observations. — It is often necessary in sur- 
veys of sediment-bearing streams to determine the amount of 
silt carried by the water in suspension. The work consists of 
three operations, namely: (1) obtaining the samples of water; 
(2) weighing or measuring out a specific portion of each, mix- 
ing these in sample jars according to some system, and setting 
away to settle ; (3) siphoning off the clear water, filtering, and 
weighing the sediment. Sometimes a fourth operation is re- 
quired, which is to examine the sediment by a microscope on 
a graduated glass plate, and estimate the percentages of differ- 
ent-sized grains. The sedimentary matter carried in suspen- 
sion may be divided into two general classes, — that in continu- 
ous suspension, and that in discontinuous suspension. The 
former is composed of very fine particles of clay and mud 
whose specific gravity is about unity, so that any slight dis- 
turbance of the water will prevent its deposition. This once 
taken up by a running stream is carried to its mouth or -caught 
in stagnant places by the way. The matter in discontinuous 
suspension consists of sand, more or less fine according to the 
velocity and agitation of the current. This matter is con- 
stantly falling towards the bottom and is only prevented by the 



330 



SURVEYING. 



% 



violent motions of the medium in which they are suspended. 
These particles are constantly being picked up where the ve- 
locity is greater, and dropped again where the velocity is less. 
A natural channel will therefore carry about the same per- 
centage of fine or continuous matter between 
#1 two consecutive tributaries, but of the coarser 

material there will be no uniformity whatever in 
successive sections in this same stretch of river. 
In natural channels there are always alternate 
engorged and enlarged sections for any particu- 
lar stage of river, and the positions of these en- 
gorgements and enlargements are different for 
different stages. In fact, the engorged sections 
at high water are usually the enlarged sections 
at low water, and vice versa. If the bed is fria- 
ble the engorged section is always enlarging, and 
the enlarged section is constantly filling as a^ 
result of the discontinuous movement of sedi- 
mentary matter. The cause of these relative 
changes of position of engorged and enlarged 
sections is the great variation in width.* 

It is the discontinuous sediment which is of 
principal significance to the engineer, for this 
is the material from which sand-bars are formed 
which obstruct' navigation, and it is also the ma- 
terial from which he builds his great contraction 
works behind his permeable dikes. The water 
being partially checked behind these dikes at once drops the 
heavier sediment, and so artificial banks are rapidly formed. 
The continuous sediment is of little consequence to the engi- 
neer. 



Fig. 88. 



* See paper by the author entitled " Three Problems in River Physics," be- 
fore the American Association for the Advancement of Science, Philadelphia 
meeting, 1884. 



HYDROGRAPHIC SURVEYING. 33 I 

262. Collecting the Specimens of Water. — It is neces- 
sary to take samples of water from various parts of the cross- 
section in order to obtain a fair average. Surface and bottom 
specimens should always be taken, and if great exactness is 
required specimens should also be taken at mid-depth. One 
of each of these should be taken at two or three points on the 
cross-section. A full set of specimens is collected once or 
twice a day. A special apparatus is required for obtaining 
samples from points beneath the surface. The requirements 
of such an apparatus are very well satisfied by the device 
shown in Fig. 88, which the author designed and used very 
successfully in a hydrographic survey of the Mississippi River 
at Helena, Ark., in 1879.* ^ is a galvanized iron or copper 
cup ; /an iron bar one inch square; L a mass of lead moulded 
on the bar at bottom ; B the bottom cup for bringing to 
the surface a specimen of the bottom, / being a leather cover; 
W the springing wire by which the lids a a are released and 
drawn together by the rubber bands b b when the apparatus 
strikes the bottom, or when this wire is pulled by an auxil- 
iary cord from above; d d adjustable hinges allowing a tight 
joint on the rubber packing-disks c c when the lids are closed. 
In descending, the lids are open and the water in the can C is 
always a fair sample of the water surrounding the apparatus. 
When the lids are closed the sample is brought securely to 
the surface. The can when closed should be practically water- 
tight ; if it leaks at bottom some of the heavier sediment is, 
likely to escape, for it settles very quickly. The bottom speci- 
men should be taken about a foot above the bottom to avoid 
getting an undue portion of sand which is at once stirred up 
by the apparatus striking the bottom. 

263. Measuring out the Samples. — A given portion of each 
specimen by measure or by weight is selected for deposition. 

* See Report of Chief of Engrs., U. S. A., 1879, v °l- iii- » p. 1963. 



33 2 SURVEYING. 



Great care must be exercised in obtaining the sample volume. 
It cannot be poured off, even after violent shaking, for the 
heavy sand falls rapidly to the bottom. A good way is to 
draw it from the vessel by an aperture in its side while the 
water is stirred within ; greater accuracy can be attained by 
weighing the sample of water than by measuring it. All the 
samples of a given kind are then put together in one jar, which 
is properly labelled, and set away to settle. Thus, all the sur- 
face samples are put into one jar, the mid-depth samples in 
another. The Mississippi and the. Missouri River water re- 
quires about ten days' settling to become clear. 

264. Siphoning off, Filtering, and Weighing the Sedi- 
ment. — When the water has become quite clear it is carefully 
siphoned off, and the residue is filtered through fine filter paper 
(Munktell's is best). Two papers are cut and made of exactly 
the same weight. One is used for filtering and the duplicate 
laid aside. The filter-paper containing the sediment and also 
its duplicate are then dried in an oven at a temperature not 
higher than 180 . When quite dry the sediment paper is put 
in one pan of the balance, and the duplicate in the other and 
weights added to balance. The sum of the weights is the 
weight of the sediment. This divided by the weight of the 
sample of water, usually expressed by a vulgar fraction whose 
numerator is one, is the proportionate quantity sought. 



CHAPTER XL 
MINING SURVEYING. 

265. Definitions. — Mining Surveying, like all other classes 
of surveying, has for its object the determination of the rela- 
tive positions of the different portions of the subject of the 
survey. The same principles which are employed in surveying 
on the surface govern the engineer in the prosecution of a 
mining survey. In fact, mining surveying may be considered 
as an extension of topographical surveying to the accessible 
portions beneath the surface of the earth, with certain modi- 
fications of the adjuncts of surface surveying, necessitated by 
the nature of the case. 

The parts of a mine included in a mining survey are the 
surface and surface-workings, shafts, tunnels, inclines or slopes, 
drifts, stopes, winzes, cross-cuts, levels, air-courses, entries, and 
chambers. 

Surface-workings include open cuts, pits, and other exca- 
vations of limited extent. 

A Shaft is a pit sunk from the surface more or less perpen- 
dicularly on the vein or to cut the vein. The inclination of 
the vein is called the Dip, or Pitch, and its direction across the 
country is called the Strike. 

A Tunnel is a horizontal excavation from the surface along 
the course of a vein, or across the course of known veins, for 
purposes of discovery. The approach to a tunnel is called an 
Adit. 

An Incline or Slope is a tunnel run at an angle to the hori- 
zontal. 



334 



SURVEYING. 



A Drift is a tunnel starting from an underground working 
such as a shaft. When there are a series of drifts at different 
depths, they are termed Levels ; as first level, second level, 
or 50-foot level, 100-foot level, etc. 

A Stope is the working above or below a level from which 
the ore is extracted. An overhand or back stope is the work- 
ing above a level ; an underhand stope is the working from 
the floor. 




A Winze is a shaft sunk from a leveL 

A Cross-cut is a level driven across the course of a vein. 

An Air-course is a tunnel driven for the purpose of venti- 
lation. 

An Entry is a passage through a mine. 

A Chamber is a large room from which the ore is mined. 

The last two terms are used more especially in coal-mines, 
where the vein lies flat or nearly so. 



MINING SURVEYING. 335 

The operations of a mining survey are conducted like those 
of a topographical survey. An initial point is selected usually 
from its importance to the object sought, and all the subse- 
quent stations are connected either directly or indirectly with 
it, and their positions with reference to it shown on the map 
of the survey. 

266. Stations are occupied by candles or lamps constructed 
for the purpose, in place of poles, flags, etc., as on the surface. 
An illuminated plumb-line is a good substitute for a lamp, and 
gives the observer a greater vertical range, which is helpful in 
case the station is obscured by intervening objects. 

Owing to the peculiar nature of the survey, it is imprac- 
ticable and sometimes inexpedient to mark stations as on the 
surface ; recourse is therefore had to other devices, which must 
be employed to suit circumstances. 

It would not be advisable, even if it were practicable, to 
leave a station-mark on the bottom of any portion of a mine, 
as frequent passing would disturb or obliterate it. 

It is better therefore to leave the mark overhead, if acces- 
sible and not liable to be disturbed, either by driving a nail in 
a timber should one be convenient, or by drilling a hole in 
which may be inserted a wooden plug, properly marked, or 
simply by cutting a cross or other device on the exposed sur- 
face. Another method, where the above cannot be employed, 
is by marking points on the walls and measuring the respective 
distances from the station to them. 

267. Instruments. — Steel tapes only should be used for 
measuring, being more convenient and less liable to inaccuracy 
than a chain. These may be of different lengths to suit the 
work on which they are to be employed ; sometimes, as in the 
case of coal-mine surveys, tapes of several hundred feet in 
length can be employed to advantage. 

The Compass, unless used as an angular instrument to de- 
flect from an established line, should not be employed in min- 



336 



SUR VE YING. 



ing surveys, as the variation between stations is so inconstant 
as to render it unreliable when used to deflect from the mag- 
netic meridian. The magnetic needle may be used, however, 
in connection with the transit as a check. 

The Transit alone should be used in important work, and 
certain additions to it for vertical pointings will be found indis- 
pensable. In sighting up or down a shaft the 
ordinary form becomes useless when the line 
of sight passes inside the upper plate of the 
instrument. A prismatic eye-piece, Fig. 90, 
will overcome this difficulty for upward sights, 
but the survey cannot be carried downward by 
its use.. 

An extra telescope attached either to the top or side of the 
central telescope will overcome this difficulty. The attach- 
ment to the top is made as shown in Fig. 91 by coupling-nuts, 




Fig. 90. 




Fig. 91. 



which fasten it firmly over the centre of the instrument. The 
attachment to the side, Fig. 92, is effected by means of a spindle 
from the attached telescope which fits into the hollow axis of 
the central telescope and is secured by means of a clip which 



MINING SURVEYING. 



337 



passes through both the axis and spindle. A counterpoise is 
similarly attached to the opposite side of the central telescope 
to preserve the equilibrium. 

This latter form of attachment is more com- 
pact than the former, the principal objection to 
it being that a correction must be applied to 
each reading of a horizontal angle equal to the 
tangent of an angle whose opposite side is the 
distance between the centres of the telescopes, 
and whose adjacent side is the horizontal dis- 
tance between stations. This objection, how- 






Fig. 92. 



Fig, 93. 



ever, is removed by a simple device. Two brass tubes, Fig. 93, 
about two inches long, are connected by an intermediate web, 
so that the distance between their centres shall exactly equal 
that between the centres of the telescopes. One of the tubes 
is of sufficient size to enclose a pike-staff graduated to fractions 
of a foot, upon which it can be easily moved to any desired 



22 



33^ SURVEYING. 



height, and the other large enough to contain-a candle, and has 
a light plumb-bob suspended below its centre the better to 
maintain the staff in a perpendicular position ; the staff now- 
being placed over any station and the brass tube and candle 
set at the height of the instrument on the staff, which is held 
in a perpendicular position with the line between the tubes 
parallel to the horizontal axis of the telescope, a reading can 
be made to the flame of the candle which gives at once the 
true azimuth of the line and the dip of the shaft. 

In using this device the side telescope and tube carrying 
the candle should always be on the same side of the line. The 
transit must always be placed exactly over the point occupied 
by the foot of the staff ; and here it may be well to state that 
the greatest care and accuracy must be exercised in exactly 
centring the instrument over the station, as the courses are 
carried forward entirely by deflection angles, so that an error 
introduced at one station is carried through all and increased 
at each. Again, in sighting down a shaft, although the per- 
pendicular distance may be considerable, the horizontal dis- 
tance between stations must be small, so that even a slight 
error made in a shaft will be of considerable magnitude when 
carried out in the levels of a large mine. 

The transit should have an extension tripod, Fig. 94, so 
that one or more of the legs can be shortened, the better to 
place it over a station on a steep mountain-side or in a mine, 
or to lower the instrument to see under intervening objects, or 
to adapt it to different heights of the workings of the mine. 

The Mining Transit should be provided with the Solar At- 
tachment, that all lines of the survey may be referred to the 
meridian. 

In unimportant surveys the pitch of the shaft or the dip of 
the vein may be determined by the clinometer or by measuring 
the horizontal and perpendicular distances between any two 
conveniently located points of the foot or hanging wall and 



MINING SURVEYING. 



339 



calculating the pitch or dip from the measurements thus ob- 
tained. 

A plummet-lamp, Fig. 95, will also be found very convenient. 




Fig. 94. 



Fig. 95. 



268. Mining Claims. — The first work of the surveyor upon 
a mining claim is its location. Mining claims are of different 
dimensions according to the local laws and customs of the 



340 SUR VE YING. 



country; varying from 50 feet to 600 feet in width and from 
100 to 3000 feet in length. In the earliest days of Western 
mining, the dimensions of a claim were decided at a convention 
of all the miners in the district. Now the United States laws 
limit the length to 1500 feet, but the width still varies not 
only in different States, but in different counties in the same 
State. 

The form of a mining claim is essentially a parallelpgram, 
being regulated by the U. S. mining laws, which prescribe that 
whatever the relative position of the side lines to each other, 
the end lines must be parallel. 

This is to prevent more than fifteen hundred feet of a vein 
or lode from being included in one claim. The side lines of a 
claim may be straight lines extending between the ends of op- 
posite end lines, or they may be broken lines to include the 
vein if it should be curved, so as to pass outside straight lines ; 
but in any case they can only include 1500 feet of the vein 
measured along the centre line of the claim. A mining claim 
is included between parallel vertical planes passed through the 
end lines ; but a miner has a right to follow his vein downward, 
although it so far passes from the perpendicular in its down- 
ward course as to extend beyond vertical planes passed 
through its side lines. 

The above are the essential features which govern the shape 
of a mining claim. 

The method of procedure in making the location is as fol- 
lows: When the discoverer of a mine has sunk his shaft ac- 
cording to law, so as to expose 10 feet of the vein, he is entitled 
to have his claim surveyed and recorded. He then decides 
how much of the 1 500 feet he desires to extend on either side of 
his discovery-shaft along the vein. He is governed in this 
by various considerations, such as his proximity to other claims, 
the promise of mineral in different portions of the lode, or the 
nature of the ground. The surveyor begins his survey for 



34i 



COPY OF AN ACTUAL PLAT OF 
PATENTED LOOE8 IN 

Sugar Loaf Mining District, 

BOULDER COUNTY, COLO. 

C. A. Russell, 

U. S. Deputy Mineral Surveyor. 




Fig. 96. 



34 2 SURVEYING. 



location at the point of discovery, and runs from it in opposite 
directions until he has measured off 1500 feet. 

The survey has thus far been run on the centre line of the 
claim. On arriving at the ends of the line, the surveyor 
measures off half the width of the claim on each side of the 
centre line, generally at right angles to it if the claim is 
straight, and sets his corners at the ends of the end lines. He 
also places monuments on the side lines, midway between the 
corners, called the side-line centre monuments, the law re- 
quiring that the claim shall be distinctly marked upon the 
ground, so that its boundaries can be readily traced. This 
much of the survey being now completed, it remains to run a 
tie line from some corner of the claim to a well-known monu- 
ment. This must be a section corner of the Government 
surveys if the claim be on surveyed lands, otherwise to a 
prominent natural object, or to a locating monument estab- 
lished for the purpose. This is done to identify and locate 
the claim so that its locus may be a matter of record. An 
example of an actual plat of mining claims is shown in Fig. 96. 
The next survey of a mining claim is its survey for patent of 
the United States. The original location may be made by 
any surveyor, and is sometimes made by the miner himself; 
but the survey on which the patent or title from the United 
States is issued must be made by a deputy of the U. S. 
Surveyor-general of the Public Lands, who is thus known as a 
U. S. Deputy-mineral-surveyor. These officers give bonds to 
the Government in the sum of $10,000 for the faithful perform- 
ance of their work, and are required to pass an examination, 
that the Surveyor-general may be satisfied of their capability. 

The survey for patent must be made with the greatest care 
and accuracy. It must exactly locate the claim with reference 
to a corner of the public surveys, if such be within two miles, 
and must show the nature and extent of the conflict with 
other official surveys if it should conflict with any, or with 



MINING SURVEYING. 343 

other mining claims not officially surveyed if it is desired to 
exclude from the claim the area in conflict. 

A specimen of the field-notes of a survey for patent issued 
for the instruction of the U. S. Deputy-surveyors of Colorado 
is given in Appendix B. 

The Surveyors-general of the different States and Territo- 
ries issue instructions to their deputies, and these, with a knowl- 
edge of the U. S. mining laws, must govern the surveyor in his 
work; but as they are more strictly legal than mathematical, it 
is not important to consider them in this chapter.* 

The foregoing surveys are strictly land surveys, and are 
only mentioned to illustrate the method of staking out a min- 
ing claim and to give some idea of the shape and size. 

UNDERGROUND SURVEYS. 

269. Mining Surveying proper, or the underground work 
of the survey, will be considered in a few practical examples 
selected from actual cases. 

270. To determine the Position of the End or Breast of 
a Tunnel and its Depth below the Surface at that Point. — 
Set up the instrument at a point outside the tunnel, so as to 
command as long a sight as possible into the tunnel and also 
the surface of the mountain above it. If the end of the tun- 
nel can be seen from the station a course and distance can be 
taken at once to the breast, and this course and distance dupli- 
cated on the surface. Vertical angles can then be measured 
to the points thus determined on the surface and in the tun- 
nel, and the calculation of the depth of the breast below the 
surface may be made from the data thus obtained. 

* Copies of the Instructions can be procured of any Surveyor-general on 
application. Those for Colorado are given in Appendix B. The U. S. mining 
laws, together with all State and Territorial laws and local mining rules and 
regulations, are compiled in vol. xiv. of the U. S. Census for 1880, 4to, 705 pp.. 
1885. This is a most valuable publication. Price $4 if not obtained through 
an M. C. 



i 



344 



SURVEYING. 



In case the breast is not visible from the first station, take 
as long a sight as practicable to Station No. 2, and before re- 
moving the instrument reproduce Station No. 2 upon the sur- 




N<$ 



RLA'N OFTUNNELAND SHAFT 

Scale lin.=100 ft, 
Fig. 97. 

face as in the preceding case, thus avoiding a resetting of the 
instrument at Station No. 1 when the underground work is 
completed. At the same time measure the vertical angle to 
Station No. 2 in the tunnel. Set the instrument at Station 
No. 2, and, after having obtained the back readings to Station 
No. 1, measure the course, distance, and vertical angle to Sta- 
tion No. 3. 

Repeat the above operations at the different stations until 






MINING SURVEYING. 



345 



the breast is reached, taking any measurements of the dimen- 
sions of the work that may be necessary, and leaving station 
marks for future reference, as described in article 266. Set the 
instrument over Station No. 2 on the surface and very care- 
fully duplicate the courses and distances measured in the tun- 
nel, at the same time noting the vertical angles between the 
surface stations. The vertical angles can be measured most 
easily by sighting to a point on a short staff at a height above 
the station equal to the height of the instrument. 

It is advisable to explore the tunnel before surveying it, as 
then any difficulties can be provided for and the stations 
selected more advantageously. Sometimes the course from 
Station No. I to Station No. 2 is assumed as a meridian of the 
survey and all courses deflected from it, but it is better to use 
the true solar course between these stations because the field 
notes can then be placed in the table for calculation without 



further reduction. 



Example. — Following is a specimen of field-notes of the survey of a tunnel 
both underground and surface: 

FIELD-NOTES. 



Station. 


Vertical Angles 


Course. 


Distance. 


Remarks. 








in Tunnel. 


on 
Surface. 








I 






S. 36 50' W. 
S. 36 50' w. 


19. 1 ft. 
99.I 


to mouth of tunnel 




-f 1° 18' 


+io° 35' 


to Sta. No. 2. 


2 


+ o° 31' 


+15° 43' 


S. 49 47' W. 


104.2 


" 3- 


3 


+ o° 45' 


+14° 27' 


S. 40 0' W. 


37-1 


4' 


4 


- o° 34' 


+16 17' 


S. 4°55'E. 


56.5 


" 5. 


5 


+ 3° 37' 


+12° 21' 


S. 71 15' E. 


46.0 


" 6. 


6 


+ 3° 3o' 


+13° 56' 


S. 77° 30' E. 


40.7 


to breast of tunnel. 


Breast. . 




-17° 56' 


N. 16 16' E. 


266.57 


from station on surface 
over breast of tunnel to 
Sta. No. 1. 



346 



SURVEYING. 



The following table shows the method of reducing the survey. The first 
six columns represent the ordinary method of reducing a traverse to a straight 
line. The agreement between the resultant and the check course proves the 
accuracy of the field-work. 

Columns 7 and 8 contain the vertical angles in the tunnel and the rise or 
fall in feet corresponding to them. 

Columns 9 and 10 similarly contain the vertical angles of the courses and 
distances on the surface and the difference of elevations between stations corre- 
sponding to them. 

The algebraic sum of the vertical heights in the tunnel gives the difference 
of level between the Station No. 1 and the breast ; and the sum of the differ- 
ences of elevation in column 10 gives the total difference of elevation between 
Station No. 1 and the point on the surface over the breast. 

The difference of columns 8 and 10 shows the depth of the breast of the 
tunnel below the surface. The tangent of the angle obtained by dividing the 
sum of the elevations in column 10 by the length of the resultant distance 
should agree with the vertical angle read to Station No. 1 from the point on 
the surface over the breast of the tunnel. 

The following is the form used in reducing the field-notes: 



OFFICE FORM. 



Course. 



S. 36 50' W. 
S. 49° 4/ W. 
S. 40 oo' W. 
S. 4°55'E. 
S. 71 15' E. 
S. 77° 3o' E. 
Resultant course, 
N. 16 20' E. 



Dist. 



99.1 
104.2 
37- 1 
56.5 
46 
40.7 

265.62 



Latitude. 



N. 



254.90 



79 -3 2 
67.28 
28.42 
56.29 
14.78 



Departure, 



E. 



4.84 
43-56 
39-73 

74.69 



W. 



59 -4 1 
79-57 
23.84 



254.90 254.90 162.82 162.82 
Check: 85.56 -*- 265.62 = 0.3221 = tani7° 51'. 



Vert. 
Angle. 



+ i° 18' 

+ o° 31' 

+ o* 45' 
- o° 34' 
+ 3° 37' 
+ 3° 3o' 



Rise or 

Fall. 



+ 2.24 

+ °-94 
+ 0.48 
— 0.56 
+ 2.91 
+ 2.49 



Total + 8.50 



Vert. 
Angle. 



+ io° 35' 

+ i5° 43' 

+ 14° 27' 

+ 16 17' 

4- 12 21' 

+ 13° 56' 



Rise or 

Fall. 



+ 18.51 
+ 29.32 
+ 9-56 
-f 16.50 
-f- 10.07 
-f- 10.10 



Total + 94.06 
8.50 



Depth below surface = 85.56 



271. Required, the Distance that a Tunnel will have to 
be driven to cut a Vein with a Certain Dip. — Case I. When 
the direction of the tunnel is at right angles to the course ', or par- 
allel to the pitch of the vein. The dip having been first ascer- 



MINING SURVEYING. 



347 



tained by sighting down a shaft sunk on the vein, or by any 
other practicable method, set up the instrument on the apex 
or outcrop of the vein directly over the line of the proposed 
tunnel and measure the vertical angle and horizontal distance 
to the mouth of the tunnel. 

From the results obtained calculate the depth at which the 
tunnel will intersect the vein, then from this depth and the 
angle of the dip, calculate the horizontal distance of the vein 
from a vertical line through the instrument station. 

This distance, added to or subtracted from the horizontal 
distance between the station and mouth of the tunnel, accord- 
ing as the dip is from or toward the mouth, will give the re- 
quired distance. 




POINT OF INTERSECTION OF VEIN AND RUNNEL 



"#" 






j|F 


OUTCROP OF VEIN! 


JN.45 E.. 




1 
1 
i 
i 

i 
i 
1 
1 
1 
r 


i 

i 

i 
1 




?! 


8 




*■ 


139 




V 


1(0 




Oi 


|m 




3*i 


o 




IA i 


m 




8; 


H 
jc 




«• 1 


|Z 




1 
I 

1 


IP 

l 




1 
1 

1 
1 
1 

1 
1 
1 
1 


i 

1 
1 
i 
i 

j 

i 
i 
I 

1 



Fig. 98. 

Example. — A certain vein has a course of N. 45 E. and its pitch is N. 
45° W. with a dip of 15 from a vertical line. The horizontal distance to the 
mouth of a cross-cut tunnel from the apex of the vein is 200 feet S. 45 E. and 
the vertical angle is — 25 . 

At what distance from the mouth will the tunnel intersect the vein? (Fig. 98.) 



34^ SURVEYING. 



Depth at which the tunnel will intersect the vein = 200 X tan. 25° = 93.26 
ft. 

Distance of vein from vertical line at depth of 93.26 ft. = 93.26 X tan. 15 
== 24.99 ft* 

Adding the last result to the horizontal measured distance, we have 24.99 
-j- 200 = 224.99 ft«» ^e distance from the mouth of the tunnel to its intersection 
with the vein. 

Case II. When the direction of the tunnel is oblique to the 
course of the vein. Proceed as in Case I. and measure the hori- 
zontal distance from the instrument station to the mouth of 
the tunnel, the vertical angle and the dip, also the angle which 
the course of the vein and the line of the tunnel make with 
each other. Calculate the depth at which the tunnel will in- 
tersect the vein, and the distance of the vein at the tunnel level 
from a vertical line through the station, as in the previous case. 
Multiply this distance by the cosectant of the angle between 
the courses of the vein and tunnel and apply it to the meas- 
ured horizontal distance, as in Case I., and we have the re- 
quired result. 

Example.— A certain vein has a course of N. 45 E. and its pitch is N. 45 
W. with a dip of 15 from a vertical line. The horizontal distance to the 
mouth of a cross-cut tunnel running due west is 200 ft. due east and the 
vertical angle is — 25 . 

At what distance from the mouth will the tunnel intersect the vein ? (Fig. 99.) 

Depth at which the tunnel will intersect the vein = 200 X tan. 25 = 93.26 ft. 

Distance of vein from vertical line at depth of 93.26 ft. = 24.99 ft* 

Angle between course of vein and line of tunnel = 45 . 

Multiplying, 24.99 X cosec. 45 = 35.34 ft. Add the result to the horizontal 
measured distance and we have 2004-35.34 = 235.34 ^-> the required distance 
from the mouth of the tunnel to its intersection with the vein. 

272. Required, the Direction and Distance from the 
Breast of a Tunnel to a Shaft, and the Depth at which it 
will cut the Shaft. — Make a survey of the tunnel and repro- 
duce it upon the surface, as in the first example. Calculate 
the depth of the breast below the surface. Set up the instru- 



MINING SURVEYING. 



349 



ment at the shaft and measure the vertical angle and hori- 
zontal distance to the point on the surface over the breast. 
Calculate their difference of level from the measurements ob- 



^ 






& 



.°v 










tained and add it to or subtract it from the depth of the 
breast below the surface. The result is the depth of tunnel 
below the mouth of the shaft. 

Survey the shaft to a point whose vertical depth is equal 



350 



SURVEYING. 



to the depth of the tunnel level. . Calculate the horizontal 
distance and direction of this point from the instrument 
station at the mouth of the shaft, and mark its position upon 
the surface. Connect it with the point marking the position 
of the breast of the tunnel, and we have the line required. 
From the information thus obtained range-lines can be sus- 
pended in the tunnel, to give the direction of the shaft from 
the breast. 



Example. — A shaft whose centre at the surface bears S. 65 E. 73 ft., verti- 
cal angle -f- io° 20', from a point on the surface over the breast of the tunnel 
in the first example, has a pitch of 14 30' N. 2° 15' W. from a vertical line. 
At what direction and distance from the breast of the tunnel will it cut the 
shaft, and at what depth ? (Fig. 97.) 

The following is the form of the field-notes: 

FIELD-NOTES. 





Vektical 


Angles. 








Station. 






Course. 


Distance. 


Remarks. 


In 
Tunnel. 


On 

Surface. 


1 






S. 36 50' W. 


19. 1 


To mouth of tunnel. 




+ i° 18' 


+ io° 35' 


; S. 36 50' w. 


99.1 


" Station No. 2. 


2 


+ o° 31' 


+ i5° 43' 


S. 49° 47' W. 


104.2 


.. » « 3> 


3 


+ o° 45' 


+ 14° 27' 


S. 40 00' w. 


37-1 


it (4 IS 

4- 


4 


- o° 34' 


+ 16 17' 


S. 4°55'E. 


56. 5 


" " " 5- 


5 


+ 3° 37' 


-|- 12° 2l' 


S. 71° 15' E. 


46.0 


" " " 6. 


6 


+ 3° 3o' 


4- 13 56' 


S. 77 30' E. 


40.7 


" breast of tunnel. 


Breast. 




-f- IO° 2o' 


S. 65 00' E. 


73 


" centre of shaft. 


Centre of 


In shaft. 






In shaft. 




Shaft. 


- 75° 3o' 




N. 2 i 5 ' W. 


102.12 


" point in shaft at tunnel 
level. 



The depth of the tunnel at the breast is determined as in the first example. 
The vertical distance between the point over the breast and the mouth of the 
shaft is determined by the equation: 

Difference of elevation = 73 X tan io° 20' = 13.31 ft. 

Add this to the depth of the tunnel at the breast and we have 13.31 -f- 85.56 = 
98.87 ft., the vertical depth at which the tunnel will cut the shaft. With this 



MINING SURVEYING. 35 1 

depth and the pitch of the shaft, 14° 30', we obtain the depth (distance along the 
shaft) at which the tunnel will cut the shaft, measured along the dip, and also 
the horizontal distance from the instrument station, which is the intersection 
of the centre line of the shaft with the surface, to the point of intersection, by 
the following equations: 

Depth measured on the dip = 98.87 X sec 14 30' = 102.12 ft. 
Horizontal distance = 98.87 X tan 14 30' = 25.57 ft. 

Set a stake N. 2° 15' W. 25.57 ft. from the instrument, and connect it with the 
point over the breast of the tunnel, and we have the course and distance 
from the breast of the tunnel to the line of survey down the shaft, S. 85 21' 
E. 65.22 ft. 

273. To survey a Mine, with its Shafts and Drifts. — 

Set up the instrument at the top of the main shaft, and after 
having first obtained the meridian, take the bearing-distance 
and vertical angle to the point selected for the first station in 
the shaft. The distance is to be measured on a direct line be- 
tween the stations, and its horizontal and vertical components 
afterwards calculated from the data obtained. The stations 
in the shaft are to be selected with a view to the extension of 
the survey into the different levels and down the shafts, and, 
as in case of other underground surveys, it is well to explore 
the mine ahead of the work, that the stations may be selected 
advantageously. 

The field-notes of the survey of a mine are here given for illustration. The 
horizontal and vertical components of the distances measured down the shafts 
can be obtained by the use of a table of natural sines and cosines. (Figs. 100 

and 10 1.) 



352 



SURVEYING. 



FIELD-NOTES. 



Sta. 



Course. 



N. 53° 57' W. 
N. 34 52' E. 
S. 25 13' W. 

N. 57 46' W, 
N. 3 o» 30' E. 



N. 69 23' W, 
N. 28 24' E. 

S. 19 59' W. 

N. 71 15' W. 
N. 37 20' E. 

S. 28 43' W. 

N. 72 01' W. 



S. 71 so' E. 

S. 71 50' E. 

N. 66° 15' W, 
N. 32° 50' E. 

S. 24 00' W. 

N. 55° 03' W. 
N. 34 15' E. 

S. 37° 45' W. 

N. 88° 30' W. 

N. 34 00' E. 

S. 24 00' W. 



Dist. 



54-i 
57-6 
60.8 

60.0 
73-o 



47-5 
75 

90 

55 
75 

64 

55-6 



5-i 
8.1 

46.8 
60 

5i 

40 
39 

55 
48.6 

54 
15 



Vertical 
Angles. 



75" 41 
o° 00 
o° 00 

66° 24 
o°oo' 



77" 30 
o° 00' 



78°oo 
o°oo 

o°oo 
79° 30' 



- 85 00' 
o° 00 



86° 46 
o c 00 



Hori- 
zontal 

Com- 
ponent. 



J 3-37 



24.02 



10.76 



"•43 



10.13 



4.07 



2.25 



7.60 



Vertical 
Com- 
ponent. 



52.42 



54.98 



46.37 



53-So 



54-67 



46.62 



39-93 



48.0 



Remarks. 



Begin at Sta. 1 at top of shaft. 

To Sta. 2 at 1st level in shaft. 

" air-shaft at end of 1st level. 

" centre of bottom of discov- 
ery-shaft, 50 ft. deep. 

" Sta. 3 at 2d level in shaft. 

" breast of 2d level, running 
N.E. Second level run- 
ning S.W. filled with de- 
bris, not accessible. 

" Sta. 4 at 3d level in shaft. 

" breast of 3d level, running 
N.E. 

" breast of 3d level, running 
S.W. 

" Sta. 5 at 4th level in shaft. 

" breast of 4th level, running 
N.E. 

" breast of 4th level, running 
S.W. 

" Sta. 6 at 5th level in shaft. 
The vein here divides : 
the shaft follows the por- 
tion to the south. The 
shaft is chambered out at 
this point, being 10 ft. 
wide S.W., and 20 ft. long 
S.E. from Sta. 6. 

" Sta. 7 at top of shaft in 
chamber. 

" Sta. 8 opposite drift, run- 
ning S. 24 15' W., 102 ft. 

" Sta. 9 at 6th level in shaft. 

" breast of 6th level, running 

N.E. 

" breast of 6th level, running 
S.W. 

" Sta. 10 at 7th level in shaft. 

" breast of 7th level, running 

N.E. 

" breast of 7th level, running 
S.W. 

" Sta. 11 at 8th level, at bot- 
tom of shaft. 

" breast of 8th level, running 
N.E. 

" breast of 8th level, running 
S.W. 



MINING SURVEYING. 



353 



Note. — The width of the levels of this mine are about four feet. The 
dimensions of the shaft are 4' X 10'. The line of survey down the shaft was 




LONGITUDINAL 
SECTION 



transverse 
"section 



Fig. 100. 



run about 2 ft. from the north end of the shaft. In the levels the line of survey 
was to the centre of the breast. 
23 



354 



SURVEYING. 



Fig. 101 shows the plan, and Fig. ioo the longitudinal and transverse sec- 
tions of the mine as plotted from the field-notes. The plan is plotted from the 
courses and horizontal components of the measurements in the shaft and levels, 
as projected upon a horizontal plane. 

The longitudinal section is platted from the courses and vertical components 
of the measurements in the shaft, and the horizontal measurements in the 
levels as projected upon a vertical plane passing through Station i and at right 



PLAN 




angles to a vertical plane passing through Stations i and n, upon which the 
transverse section is plotted as projected from the vertical angles, courses, and 
measurements in the shaft. 

Thus it will be seen that for the full representation of an underground sur- 
vey, showing the relative position of the parts to each other, three planes are 
necessary, — two vertical planes at right angles to each other, and a horizontal 
plane. 



274. Conclusion. — The above examples comprehend some 
of the more general cases arising, in the practice of mining 



MINING SURVEYING. 355 

surveying; any other cases which may arise will be found to 
be modifications or combinations of these. The problem to be 
considered can be solved by an application of the principles 
therein embraced, which the surveyor will find useful, also, in 
solving problems of mining engineering relating to the meas- 
urement of ore reserves, development, and systems of working. 
It has been shown that the following out of the underground 
workings of a mine corresponds to traversing when elevations 
are carried by means of vertical angles, as was fully described 
in the chapter on topographical surveying. The notes are 
also reduced in the same manner. 

It has been the object of this chapter to present the subject 
of mining surveying in as simple a form as possible, and divest 
it of all features which, although they may give it a distinctive 
aspect, serve only to render it more complex and give the 
reader an idea of difficulties which are only imaginary. 

It is useless, also, for the mining surveyor to encumber him- 
self with many paraphernalia. Good work can be done with 
a mining transit provided with an extra telescope for vertical 
pointings, one or two short rods, and a reliable steel tape, all 
of which can be carried by the surveyor on horseback over the 
rough mountainous roads. Any other adjuncts can be im- 
provised or be found at any well-conducted mine, and would 
prove more burdensome than useful. 



CHAPTER XII. 
CITY SURVEYING. 

275. Land-surveying Methods inadequate in City Work. 

— The methods described in the chapter on Land-surveying 
are inadequate to the needs of the city surveyor. The value 
of the land involved in errors of work, with such a limit of er- 
ror as was there found practicable (see art. 175), is so great as 
to justify an effort to reduce this limit. Comparing the value 
of a given area of the most valuable land in large cities with 
the value of a like area of the least valuable land which a sur- 
veyor is ever called upon to measure, the ratio is more than a 
million to one. 

This view is emphasized by the manner of use. On farm 
lands the most valuable improvements are placed far within 
the boundary-lines, but the owner of the city lot is compelled 
by his straitened conditions to place the most costly part of 
his improvements on the limit-line. His neighbor's wall abuts 
against his own. The surveyor, who should retrace this line 
and make but a small difference of location, would get his 
clients and himself into trouble. Both the value of the land 
and the manner of its use demand increased care. The modi- 
fications of the methods used in land-surveying to meet the 
requirements of work in the city will be treated in this chapter. 
Much of the work described furnishes data for the solution of 
engineering problems, but the obtaining of the facts falls en- 
tirely within the definition of surveyor's work. 



CITY SURVEYING. 357 



276. The Transit is used exclusively, but the common pat- 
tern may be very materially modified with obvious advantage. 
Seeing that the magnetic needle* is never precise and seldom 
correct, it should be wholly discarded in the construction of 
the city surveyor's transit. The verniers can then be placed 
under the eye, the bubbles can be removed from the standards 
and placed upon the plate of the alidade, and the standards 
themselves can be more firmly braced. By these changes a 
steadier and more convenient instrument is secured, when the 
useless and somewhat costly appendage of a needle-box is out 
of the way. The adjustable tripod head and the levelling 
attachment are always convenient. For topographical work, 
the vertical circle, or a sector, and stadia wires are essential, 
otherwise the methods used must be primitive. The ther- 
mometer which is needed in order to make the proper correc- 
tions for temperature may be conveniently attached to one of the 
standards facing the eye-piece of the telescope. The danger 
of breaking the tube while handling the instrument may'be 
obviated by a guard sufficiently deep to protect the bulb, made 
open on the side toward the observer. 

277* The Steel Tape is generally used for measuring. The 
legal maxim that " distances govern courses," when interpreted, 
means that, using customary methods, the intersection of two 
arcs of circles, centres and radii being known, is a more definite lo- 
cation of a point than the intersection of two straight lines whose 
origin and direction are likewise known. The fact is, the inter- 
sections are not more definite. The maxim grew into authority 
when the compass was pitted against the chain. With the 
transit to define directions of courses, and the chain still to 
measure the distances, such a maxim would not have voiced 
the results of experience, but would have been sheer nonsense. 

* The needle finds its proper place where checks are not so abundant, and in 
classes of work in which a close and rapid approximation ig of more value than 
precision. 



35 8 SURVEYING. 



The ordinary chain has too many gaping links, and the brazed 
chain too many wearing surfaces, to be kept in very close ad- 
justment to standard length. Its weight is such as to make the 
" normal tension" (see p. 375) impracticable; hence the effect 
of slight variations of pull is much more marked than if the 
tape is used. Graduated wooden rods were used until i860 to 
1870. They were unwieldy when twenty feet long, and were 
still so short that the uncompensated part of their compen- 
sating errors was a matter of considerable moment. Every 
time the pin is stuck or a mark made at the forward end of the 
tape or rod, the work is a matter of skill and involves an error 
dependent on the degree of skill attained. When the measure 
is brought forward, its proper adjustment in the new position 
is a matter requiring skill. These errors are compensating, but 
the resultant is not zero. The use of the plumb-line is another 
source of compensating errors which are reduced by an increase 
of length in the measure. First, the number of applications 
varies inversely as the length of the measure ; second, using the 
rod, it was necessary to work to the bottom of ravines and gul- 
lies and then work up again ; now the long tape spans them at 
a single application. The minus errors due to imperfect align- 
ment and inaccurate levelling of the two ends have a greater 
percentage of effect when the measure is short than when it is 
long. The longer tape brings with it some other sources of 
error. When used suspended at the ends there is a minus 
error on account of the sag of the intermediate parts, and a 
plus error from elongation due to tension ; there is also expan- 
sion by heat, which produces an error which may be plus or 
minus as the temperature at the time and place is above or/ 
below that for which the tape is tested. The effect of sag 
increases very nearly as the cube of the length when the ten- 
sion is constant. When, to counteract this increase, the 
pull is made greater than a man can apply uniformly under 
all conditions — at his feet or above his head — there come 



CITY SURVEYING. 359 



irregularities from this cause. The limit of length of tape 
which it is practicable to use will be determined by the condi- 
tions of the work and should be such that the increase of 
length involves greater error than it eliminates. On account 
of convenience in keeping tally, 50-foot and 100-foot lengths 
are generally used. In a level country the 100-foot tape is pre- 
ferred. 

There are tapes made with the purpose to eliminate the 
errors which arise from the free-hand pull, the inclination of 
the tape, and the temperature. As seen by the writer, they 
carry a spring-balance marked for a pull of ten pounds, a 
bubble adjusted to the inclination of the end of the tape at 
that pull, and a thermometer graduated to such a scale that 
each division corresponds to one turn of a screw adjusting 
the whole length of the tape to the changes of temperature. 
The whole was connected by rings and swivels, eight or nine 
wearing surfaces, some of them conical, to a tape which carried 
no graduation. The effort is laudable ; but, probably on ac- 
count of the number and form of the wearing surfaces, they 
have not yet met with general favor. Further progress may 
be made in this direction. 

LAYING OUT A TOWN SITE. 

278. Provision for Growth. — Cities grow. It is very rare 
that the considerations which should have governed have been 
given any place in determining upon the plan of the original 
town. The considerations first in importance are topographi- 
cal. What are the natural lines along which business will tend 
to distribute itself? To what form of subdivision can it adapt 
itself with the least resistance ? Where is the best harbor, 
the lake or river front, or the railway line ? Ordinarily the 
land immediately adjoining such natural features is not best 
used when used as a street, but when occupied by private 



360 



SURVEYING. 



docks, or along a railway by warehouses and factories having 
switching facilities without crossing public streets. The 
streets parallel to such lines should be of ample width, 
easy grade, and continuous but not necessarily straight align- 
ment. Much of the heavy hauling will be along such streets. 
In the business part of the town the cross-streets should 
be so frequent as to make the blocks approximately square. 
In the residence portion alternate streets in one direction may 
with advantage be omitted: this saves the expense of unneces- 
sary streets, and permanently lightens the burden of taxation. 
Which fronts are on all accounts most desirable in the par- 
ticular locality will determine in which direction the blocks 
should be longest. 

279. Contour Maps. — Another phase of topography de- 
mands attention. The sites of suburban towns may generally 
be best handled by laying out streets and lot lines in conformity 
to the undulations of the ground. Additions to the city may 
also have characteristic features that can be preserved with 
advantage. For all such cases a contour map is very useful 
to one who is able to interpret it. The making of all the 
ground available, and sightly points accessible, and at the same 
time so locating the streets as to secure economical grades, — in 
short, the judicious handling of the whole subject is facili- 
tated by the study of the contour map. 

280. The Use of Angular Measurements in Subdivi- 
sions. — Shall subdivision lines be located by an angle with the 

street on which the lots front or by 
distances from the next cross-street? 
Must distances govern courses, what- 
ever methods are used ? Let us sup- 
pose, for illustration, that it is re- 
quired to locate lot 9 in the accom- 
panying sketch (Fig. 102). Suppose, 



ete- 400 *f 

1 



1 
1 
















c 


50 
















/* 


Si 


2 


3 


4 


5 


6 


7 


8 


V 


'A 
















-A 


,50^, 


" 


" 


J> 


>i 


'11 


11 


11 


50,v/ 



Fig. 102. 



farther, that it is possible to measure each of the lines ab 



CITY SURVEYING. 36 1 



and dc with a maximum error of 1 in 5000 and that the 
conditions are such as to produce, opposite errors in the 
two lines. Then, 1st, the resulting error in locating the line 
be, i.e. {ab — dc) will be -g-^ X 400 X 2 = o. 16 feet. The 
sine of the angle by which the angle A' differs from A will be 
^i|-=: .00107. Hence the change of direction on account of 
the errors in measurement is 3f minutes. 2d, the line ef 
must be distant from ab 3f X 150 feet —550 feet, in order 
that, under like conditions, if it is measured instead of dc, the 
change in direction shall not exceed one minute. Or the loca- 
tion may be made by measuring the line ab 3 or a line near to 
it where favorable conditions exist, and then repeating ba 
the same man being fore-chainman ; the principle of reversal 
is thus applied to this measurement. Then measuring A' ' = A 
and repeating the angle, reading both verniers, the error is 
brought within the maximum error in the pointing power of 
the instrument. In order to locate be from ab parallel to ad, 
two monuments marking the line ab need to be known. The 
other method requires also a monument locating the line ae. 
It thus appears that when the side-lines of lots are located 
perpendicular, or at any other known angle with the street 
upon which the lot fronts, it is susceptible of more accurate 
location than by two (front and rear) measurements, unless the 
usual limit of error can be greatly reduced. While it is not 
likely that maximum errors of opposite character will fall to- 
gether affecting the work on the same lot, it is quite as im- 
probable that the maximum error in measuring an angle 
should vitiate the work of the transit. It is probably quite as 
easy to reduce the maximum error in measuring an angle to 
half a minute as it is to keep the maximum error in measur- 
ing distances down to 1 in 10,000. 

281. Laying- out the Ground. — The work of putting the 
plan upon the ground is a very important one. This is about 
the worst possible place to do hurried and inaccurate work. 



362 SURVEYING. 



Fences or other styles of marking possession which limit the 
contour map cannot be relied upon as defining the property- 
lines. These lines must be accurately located in relation to 
the streets of the town or of the addition, in order to make 
practicable such exchanges or sales as may be necessary to ad- 
just property-lines to the new block-lines. This method is 
preferable to that which adjusts block-lines to the original 
property-lines.* 

As a framework for the whole survey an outline figure, 
generally a quadrilateral, of sufficient dimensions, and so 
placed that it can be permanently marked with monuments 
which will remain accessible when the town is built up, should 
be located with especial care. All lines should be measured, 
all angles observed, and all practicable checks introduced. 
This figure must close absolutely ; that is, the record of the 
work when completed should be mathematically consistent. 
Unreasonable errors are to be eliminated by retracing the work. 
In the adjustment which distributes the remaining errors each 
part of the work should be weighted (art. 174, Rule 2), for it 
is very rare that a land-survey is completed under such con- 
ditions that the man who does the work would be justified, 
while these conditions are fresh in his mind, in assuming that 
the probability of error is alike at all points. The angles ad- 
mit of adjustment independently of the length of the lines. 
That distribution of the angular errors wJiich reduces the errors 
of measurement to a minimum has such weight that it can 
be overruled only by the most positive evidence that the cor- 



* In some places this idea of the private interest of the proprietor, some- 
times private spite, is carried to such an extent that it would seem that each 
man's farm or garden patch was especially fitted to be a town by itself, laid 
out with utter disregard to the towns which others are in like manner laying 
out upon adjacent farms. In this practice the interests of the public for all 
time are neglected in order to secure a doubtful advantage for one. Where the 
custom prevails it is better honored in the breach than in the observance. 



CITY SURVEYING. 363 



rections so indicated cannot be the true ones. The distances 
are then adjusted to the angles so determined. The re- 
mainder of the work of the subdivision is checked upon the 
adjusted outline, reasonable errors being distributed and un- 
reasonable ones retraced. 

282. The Plat to be geometrically consistent. — The 
necessity that the recorded plat should be consistent lies in 
the use that is to be made of it. A parcel of ground de- 
scribed by reference to the plat of record should have but one 
location, not any one of two or more possible locations, as is 
the case when the plat contains errors on its face. In the 
course of years the lines of such parcels will be retraced proba- 
bly many times, at one time by one method, at another time 
by another equally in accord with the plat. If the plat is not 
consistent with itself and with the monuments upon the 
ground, this error will be pretty sure to find its way into the 
lot location. When the fault is with the plat, it matters not 
how the monuments are placed upon the ground ; they cannot 
mark the chief points and all agree in such a way that if any 
two remain and the others are lost the relocation will in every 
case be the same. But this is just what the plat is for — to 
make a public record of the relation of each part of the sub- 
division to every other. 

283. Monuments. — How many monuments shall be lo- 
cated, and where shall they be placed ? What material shall 
be used and how set ? Answering the first question, it is plain 
that no more work should be attempted than can be done well. 
Better one point and an azimuth than points everywhere and 
no two agreeing either in distance or direction with the rela- 
tion described by the plat. But so much should be done well 
that the labor of locating any point in the subdivision from 
existing monuments shall not be excessive. The points 
chosen for placing monuments should be such as will continue 
to be accessible and will not be ambiguous. The centre lines 



364 SURVEYING. 



of intersecting streets are sometimes marked, giving one monu- 
ment to each intersection ; others choose the side-lines, giving 
four monuments to each intersection of streets. If the blocks 
are so long that intermediate points are desirable, points on 
the ridges should be selected. 

Stone is more often chosen than any other material ; iron 
bars, gun-barrels, gas-pipe, etc., are sometimes used, driven 
with a sledge ; cedar posts, say 4" X 4", are quite durable, and 
hard-burned pottery is sometimes used. Whatever material 
is chosen, the foundation, which should be flat — not pointed — 
must reach below frost; and the centre of gravity is kept as low 
as possible, so that there shall be no tendency to settle out of 
place when the ground is soft in the spring. When the tops 
are much above the surface of the ground, there is a liability 
that they may be displaced by traffic. Probably the surveyor 
does not see any traffic, or the prospect of it, when he is doing 
his work, but the traffic must come before the work of the 
monument can be spared. It is better to bury the stone wholly 
and indicate where to dig for it by bearings than to run the 
risk of losing the whole work through indiscretion in placing 
the monument that marks it. In situations where every rain 
storm produces a slight fill it is safe to place the top consider- 
ably higher than would otherwise be reasonable. The stones 
to be set are so placed in the excavation, with the heavy end 
down, that when the top is in the proper position and before any 
earth is refilled there is no tendency to fall in any direction ; then 
while the earth is being refilled and thoroughly tamped about 
the stone, the top is kept in place. It is better that the mark 
denoting the point for which the stone stands should be cut 
upon before it is placed in the ground. When this is done, if 
the mark is worn off by traffic or knocked off by accident, the 
centre of that portion of the stone which remains is a very 
close approximation to the original point. A slovenly way of 
slighting this work is to tumble the stone into the excavation, 



CITY SURVEYING. 365 



fill around it pretty much as it happens, push it to one side or 
another so that the point will come somewhere on the top, and 
then cut the mark wherever the point comes. Stones set in this 
way are liable to settle out of place after the first heavy rain, 
while frost and rain keep up their work till the stone lies flat 
upon its side. If by chance it should keep its place pretty 
well and the mark becomes defaced, it might as well be any 
loose bit of rock as a set stone, for its centre gives no idea of 
where the mark was placed. No one should be trusted to set 
corner-stones unwatched who is not familiar with the work 
and thoroughly reliable. 

Points are preserved temporarily by wooden stakes driven 
flush with the ground. The point, preserved by offsets while 
the stake is being driven, is marked by a nail. Witness-stakes 
driven alongside and standing above grass and weeds assist in 
finding the stakes when wanted. Made of half-decayed soft 
wood, e.g., old fence-boards, such stakes will hardly last a 
season ; while durable wood, well seasoned, will last much 
longer than any driven stake can be relied upon, since it does 
not go below frost, and is liable to be pushed by a passing 
wheel or be otherwise disturbed when the ground is soft. 

284. Surveys for Subdivision. — The purpose of making a 
survey before recording a plat of a subdivision is twofold, — 
first, to get the information which it is desirable to record ; 
second, to leave such monuments as will make it easy to locate 
any portion when desired. The recorded plat should show 
sufficient facts to determine the relations of every part to the 
whole, and these relations should be shown by methods which 
involve the minimum of error, i.e., giving a location which may 
be retraced with least possible doubt. The current practice 
falls short of this standard at some points which are worthy of 
note. 

(a) Surveyors seem to have no doubt of the ability of their 
field-hands to measure a line, but very seriously doubt their 



366 SURVEYING. 



own ability to measure an angle. Angles are measured dur- 
ing the progress of the work and are used for determining the 
lengths of lines ; these lengths are then made a part of the 
record, while the angles which determined them are omitted. 
Apparently some things which are dependent have become 
more certain and fixed than that upon which they depend. A 
proper record of angles would show what lines are straight and 
where deflctions are made. Defleections which are sufficient 
to very seriously affect the position of a brick wall do not show 
on the scale of the recorded plat. For example, an addition to 
a town extends from Fifth Street to Twelfth Street ; extreme 
points are well established, but intermediate monuments are 
missing ; and it is required to establish at Eighth Street the 
line of a street which a ruler applied to the recorded plat sug- 
gests is a straight line. Custom approves that in such a case 
the surveyor should try a straight line, there being a mild pre- 
sumption in its favor ; but if his straight line agrees with one 
wall and disagrees with two walls and a fence, he had better 
look further before he comes to a decision. No such doubt 
could have existed if the recorded plat had been properly made. 

(b) Very few recorded plats show what stones have been 
set by the surveyor, or indeed indicate that he has any knowl- 
edge that such monuments may ever be useful. If the custom 
were once established of noting upon the record the location 
and description of monuments, any monument found during a 
resurvey, but not shown on the record, would be discredited. 
As matters now stand it must be proved incorrect to be dis- 
credited — a thing not always easy, for a system of quadrilat- 
eral blocks whose angles are not recorded and whose street 
lines are not necessarily straight is not theoretically very rigid. 

(c) Many plats require measurements to be made along 
lines which are easily measured while the land is vacant, but 
which will become inaccessible as soon as the property is built 
up. The obstacles to be overcome before the result can be 



CITY SURVEYING. 



367 



reached by the method described on the record will each add 
to the doubt of the accuracy of that result. There are many 
ways in which plats are made, which are all justly subject to 
this criticism. Two examples will suffice. Irregularly shaped 
blocks are sometimes treated as in the annexed sketch, Fig. 
103. The outline is subdivided mechanically, and proportional 




distances are given on interior lines which are not consistent 
with any trigonometrical relation of the exterior lines, much 
less with that which does exist but is not recorded. The point 
x has nine distinct locations directly from the plat. On the 
theory that ab and cd are straight lines, their intersection gives 
one; ab straight, the distances ax and bx give each one; cd 
straight, the distances ex and dx give two. Combine the dis- 
tances ax and ex, bx and ex, etc., and get four more. But this 
is not all, for the point x stands related to each of the ten other 
points along the line ab, and each of these has also nine loca- 
tions which accord with the plat, and our point x may be lo- 
cated from either of them or any combination of them when 
they have been located by any of the methods described. 

Besides the difficulty of determining how interior points 
should be located, this fan-like subdivision wastes ground in 
each lot which results in wedge-shaped remnants about the build- 
ings, which remnants would be valuable if thrown together into 
the corners, thus keeping the remaining lots rectangular at the 



3 68 



SURVEYING. 



front. The attempt to reach a rectangular front sometimes 
fails through inattention to very simple matters, as in Fig. 104. 
Here no angles are recorded. The rear corners of the lots are 
located along the line ab by distances from aoxb; but the 
record-depths do not fall upon a straight line. The line ab 
should bisect the angle between the block-lines or be parallel 
to such bisection in order that with a constant distance along 
ab common to the series of lots on each side of that line their 



V 



120 



\s 



200 



18 



<$$\ 


\ 
3 \ 


4 \ 


6 \ 


""' V 


— »» 


1 V. 

-r^03 


2 \ 




•""""» 










17 


16 


15 


fl 


IS 


12 


11 

1 


100 


» 


j» 


» 


" 


» 



10 



100 



Fig. 104. 



angles with their respective fronts may remain constant. In 
the case given every lot-line has an angle with the block-line 
upon which it fronts different from that of every other lot-line, 
and all dependent on some block-angle which is not recorded. 
If it is not desirable to bisect the block by the line ab, its di- 
rection may be chosen as desired, the distances along it are 
fixed by the fronts on one and the angular divergence from 
that side which is chosen, and the lot fronts on the other side 
of the block must be correspondingly increased or diminished. 
When alleys are laid out in a block so that the interior lines 
are accessible, it is very rare that after the block is improved 
these lines can be measured under the same conditions as the 
fronts. If alleys are not laid out, the difficulties are usually 
much greater. Location of lot-lines by angle from the front is 



CITY SURVEYING. 369 



undoubtedly the most uniform and workmanlike method avail- 
able to the surveyor. Hence, distances on the rear lines of the 
corner lots should be omitted from the record, if their presence 
would leave any doubt as to which method of location is in- 
tended. It is not customary, nor is it desirable, that lot-lines or 
distances should be determined upon the ground before record- 
ing a subdivision, but they should be platted by a man who 
knows at least the first principles of trigonometry, and has an 
accurately measured basis for his work. 

285. The Datum-plane. — Levels referred to a permanent 
datum are needed as soon as it is apparent that the town is to 
be a living reality and not simply a town on paper. The da- 
tum is a matter of some importance, and should have a simple 
relation to some natural feature of the locality which will re- 
tain a vital interest so long as the town exists. There is an 
individuality in town-sites which usually determines for each 
case very definitely what is best. High-water mark indicating 
the danger of overflow; the lowest available outlet for a 
drainage system in a flat country ; the average sea- or lake- 
level, as affecting commerce ; these are often chosen and may 
serve as examples. The datum selected has its value accu- 
rately determined and marked by a monument as enduring 
as the granite hills, or, if that is impossible, as near this stand- 
ard as can be secured ; a block of masonry, with a single and 
durable cap-stone firmly bolted to its place, and bearing the 
datum, or a known relation to it, definitely marked and secured 
from abrasion is certainly possible for all. 

286. The Location of Streets for which the most econom- 
ical and practical system of grades may be secured is to be 
considered when the plat is being prepared. Grades are usu- 
ally established from profiles taken along the centre lines of the 
street to be graded. This method is direct and protects the 
public fund, for the grade, .which can be executed at minimum 
cost, the street being considered by itself, can be determined 

24 



370 SURVEYING. 



from such a profile. The method fails from the fact that it 
treats the fund raised by taxation as the sum total of the pub- 
lic interest. Parties representing abutting property appear 
before the legislative body which has final action and seek to 
amend the recommendation of the engineer, claiming that in- 
terests which should receive consideration are injured by the 
grades proposed. It seems plain that whatever is recommend- 
ed by the city's officer should have the moral weight which 
attaches to an impartial consideration of all the interests which 
the city fathers are bound to recognize. But this involves a 
change of method. The contour map of the district involved 
seems to offer some help toward a solution. Methods by 
which a rapid approximation of the amount of cut and fill in- 
volved in any proposed grade may be arrived at are discussed 
in Chapter XIII., on the Measurement of Volumes. 

287. Sewer Systems. — A well-devised sewer system 
touches very closely the public health. The information 
which is necessary in order to act intelligently involves, if 
storm-water is to be provided for, the area and slopes of the 
whole drainage-basin in which lies the area to be sewered. 
This will enable a close approximation to be made of the work 
required of the mains at the point of discharge. Each sub- 
district involves its own problem. The most economical 
method of reaching every point where drainage is necessary 
is learned by studying the details of topography. Borings 
along the lines of proposed work to determine the character of 
the soil and the depth of the bed-rock are necessary in order 
that contractors may bid intelligently. This species of under- 
ground topography sometimes modifies the location fixed by 
surface indications. 

288. Water-supply.— The need of a water-supply fur- 
nishes new work to the surveyor. The distance and elevation 
of the source of supply, the topography of the country through 
which aqueducts or mains must be brought, eligible sites for 



CITY SURVEYING. 37 1 



reservoirs, with their relation in distance and elevation to all 
points to be supplied, are to be furnished to the hydraulic 
engineer. The datum-plane for these maps and that of the 
town should correspond. 

289. The Contour Map, which is so generally useful from 
the time the town is first planned until public improvements 
cease to be considered, if surveyed carefully at first, has no 
need to be retraced each time such a map is useful. It had 
best be drawn in sections of sufficient scale for a working-plan, 
and so arranged that when adjacent sections are placed side 
by side the contour lines will be continuous. If the contours 
of the natural surface are drawn in india-ink, and the contours 
showing the changes made by different kinds of public work 
be drawn in some color, the map may give a great amount of 
information without becoming confused. 

METHODS OF MEASUREMENT. 

290. The Retracing of Lines comes with the private use 
of lots or blocks and with the execution of public improve- 
ments. The demand for this class of work comes not once 
only, but many times, and never ceases while there is life and 
growth. The changes to which these forces give rise furnish 
the main demand for knowing along what lines growth may 
proceed unchallenged. The man who first fences a lot in the 
middle of an unimproved block can ill afford to risk being com- 
pelled to move his fence for what a survey would cost. But 
the first attempt to go over any part of a subdivision and 
locate a lot-line raises the question, how nearly alike can a 
surveyor measure the same distance twice, or how nearly alike 
can two surveyors measure the same distance. If the distance 
noted on the recorded plat was not measured correctly, the 
resurvey must differ from it, or by chance make a mistake of 
the same amount. The difference which appears by compar- 



37 2 SURVEYING. 



ing results is not the error which exists in either the original 
or the resurvey ; it may be more than either error, it may be 
less, being the algebraic difference of the two errors. If there 
is no difference it means that the work is uniform, and may be 
correct, but both may also be in error a like amount. It has 
happened in the days of twenty-foot rods and in a city of con- 
siderable size that every rod used by surveyors was too long. 
The change to steel tapes has not set matters wholly right. 
If a man compares steel tapes bearing the brand of the same 
manufacturer and offered for sale in the same shop, he soon 
ceases to be surprised at a very appreciable difference in 
length. 

291. Erroneous Standards. — How long is a ten-foot pole 
or a hundred-foot tape is a pertinent and fundamental ques- 
tion. It cannot be ignored when deeds call for a distance 
from some other point, as fixing the beginning-point of the 
parcel conveyed. When the deed describes lot number — , as 
shown on the recorded plat, there is a theory in accordance 
with which uniformity is all that is required — a distribution of 
the distance between monuments in proportion to the figures 
of the record. Property is often laid out with a view to this 
theory of surveying. So long as block-boundaries are definitely 
marked, a degree of precision is very readily secured by this 
method which is rarely attained when surveyors attempt to 
measure standard distances. If the surveyor faithfully meas- 
ures the block through and every time distributes what he 
finds in proportion to the record, though his block distances 
may not agree with the record or with themselves, the lot-lines 
will be much more likely to be the same than if he measures 
his record distance and stops at the lot. This method assumes 
that the lots abut one upon another, and reach from one monu- 
ment to the other. But if this be true, the distances noted 
must often refer to some empirical standard peculiar to this 
block and not to the United States standard established by 



CITY SURVEYING. 373 



law. But the courts recognize no standard, so far as the 
author knows, but that which is established by law. So that 
when a surveyor comes to mark lot one, finds the corner of the 
block, and drives his stake by measuring from it the distance 
which the record assigns to lot one, it is hard to prove that he 
has not measured according to the subdivision, although he 
has given no thought to the distance which remains for the 
other lots. But trouble begins right here, for the theory which 
is correct for lot one cannot be very wrong for lot two ; con- 
tinue the process to lots six and eight, and give to another sur- 
veyor who has been doing the same kind of work at the other 
end of the block an order to survey lot seven. A conflict in 
this case is certain unless the surveyor who laid out the sub- 
division, and each of the others since, knew the length of his 
tape and knew how to measure. 

292. True Standards. — The Coast Survey Department of 
the U. S. furnishes at small cost rods of standard length at a 
temperature which is stamped on the rods. Using a pair of 
these so as to measure by contact, a standard test-rod of any 
desired length can be laid off and only such marks retained 
as may be desired. This test-rod should be of the same ma- 
terial as the tape to be tested, in order that it may have the 
same coefficient of expansion by heat and may not be affected 
by humidity of the atmosphere. Care being taken that the 
tape is of the same temperature as the rod, be it 30 , 90 , or 
6o°, when the test is made, then the tape is correct at the tem- 
perature at which the rod is correct, and this is known by the 
U. S. stamp, and has no reference to the temperature at the 
time of the test. In some styles of tape the ring may be 
shaped to make the necessary adjustment to standard length. 

Where and how to construct a standard rod, and how to 
care for it so that it may be permanently reliable, each indi- 
vidual had best determine for himself. It should be fastened 
in its place in such a manner that it can expand and contract 



374 SUR VE YING. 



freely, i.e., without any strain from its supports. If it is made 
of separate parts, these should be so joined together that there 
can be no lost motion between the pieces. The whole requires 
protection from the weather and to be so supported that it 
cannot be bent by a blow. The writer has solved this problem 
for himself in the following way : Bars of tool steel one inch 
wide and one fourth of an inch thick are joined, as shown in 
the sketch, to make the desired length 50 feet -\-; the whole is 



CT> 



Jz 



Fig. 105. 

fastened to the office floor by screws which hold the middle 
firmly, but each side of the middle the holes drilled for the 
screws are slotted sufficiently to allow for any possible change 
of temperature. The joints are so close that a light blow is 
necessary to bring the parts to place; the screws were set 
home and then withdrawn a little, so that they should not 
cause friction with the floor. After the fastening was com- 
pleted the standard marks were cut upon the rod. 

293. The Use of the Tape. — It is one thing to have a 
tape of correct length ; it is another thing to be able to use it. 
In an improved town with curb-lines clear, perhaps the most 
obvious method is by a measurement along the grade with the 
same tension as that at which the tape is tested. It is then 
necessary to correct for temperature and to note all changes of 
grade, reducing the observed distance on each grade by the 
versed sine of the inclination or by the formula given in Chap. 
XIV. By this method the tape is supported for its entire length, 
and it is practicable to use a tape two or three hundred feet 
long to advantage provided there are enough assistants to 
keep it from being broken. A difficulty arises in the use of 



CITY SURVEYING. 375 



this method from the fact that the town is not made for the 
convenience of surveyors, and curb-lines are not usually clear 
where measurements are needed, but are obstructed by piles 
of building material, bales of merchandise, etc., and in some 
towns the streets are so dirty that the graduation could not be 
seen long if a tape were used in this way ; it would also be so 
covered with drying mud that it could not be rolled in the 
box when out of use, hence would be frequently broken. 
Tapes that are wound on a reel, and have no graduations to 
speak of, could be used in the mud, but the other objections 
mentioned would still make the method of very limited appli- 
cation. It is further to be noted that the laying-out of the 
town, which is the basis of all later work, has all to be done 
before the streets are graded or the curbs set. This work 
must be done by some other method. 

The usual method is to keep the ends of the tape horizon- 
tal by using a plumb at that end of the tape where the surface 
is lowest, and often at both ends if the ground is so irregular 
' or so covered with brush and weeds that the tape must be 
kept off the ground. The tape assumes a curved form, and 
the horizontal distance is something less than the length of the 
tape. There is also a tension in the tape which, on account of 
the elasticity of the metal, somewhat increases its length. As 
the tension increases the sag diminishes, hence there is a 
degree of tension such that its effect is equal and opposite to 
the effect of the sag. Call this the normal tension. If a line is 
measured with a pull less than the normal tension for the tape 
used, the tape will sag too much and there will be a minus 
error due to this excessive sag ; if the pull used exceeds the 
normal tension, there will be a plus error due to this excess. 
It the pull has been uniform the total error in either case is 
proportional to the length of the line ; but if the pull has not 
been uniform the error has varied irregularly with each length 
of tape and can most readily be calculated by retracing the line 



376 SURVEYING. 



and using the proper tension. In practice the tape is tested 
with a known tension, and a tension so much above the " nor- 
mal " is adopted for field use that its plus error is equal to the 
plus error of the test. 

294. To determine the Normal Tension in a tape sup- 
ported at given intervals. The tape forms a catenary curve, 
since it carries no load but its own weight and is of uniform 
section. 

Let P = horizontal tension (pull) ; 

w == weight of a unit's length of tape ; 
e = base of Naperian logarithms ; 
s == length of curve from origin ; 
/ = distance between supports ; 
W — wl — weight of tape ; 
x and y = horizontal and vertical coordinates, origin at low- 
est point ; 
x = \l for cases considered. 



Then by mechanics, 



•* 



D wx wx 



2W 



p wx _ wx 

and s = — (e~P — e~ "?"). 



2W 



P 

We observe (1), that if - is constant y and s are constant for 

v J w 

the same length of tape ; (2), if P be measured, say ten pounds, 

* The discussion here given is rigid, but both the development and the evalu- 
ation of the equations are laborious. Ii the curve be assumed to be a parabola, 
which 11 may when the sag is small, the development is much simpler. See the 
ireatment of this subject in Chapter XIV. — J. B. J. 



CITY SURVEYING. 



377 



as a working condition,^ and s will vary with the weight of 

PI P 

every tape used, hence -jrr = — is the ratio which must be 

constant ; (3), if the surveyor can keep y constant, the same 
conditions keep s constant, and if y varies s must vary; (4), if 

P "WX 

x(=\l) varies, and — varies in the same ratio, then -^ is con- 

(As J^ 

stant, hence the parts of the equations in parenthesis are con- 

P 

stant and y and s vary as / and — . 



TABLES SHOWING NORMAL TENSION AND EFFECT OF 
VARIABLE TENSION. 



1 = 100 feet. x = 50 feet. 


Sag. 


Pull. 


Resultants ± 


P 
w' 


y- 


(2* - /) 
— error. 


P 

W 


Elonga- 
tion 
-f- error. 


Error in / 


Error in iooo ft. 


- 


+ 


- 


+ 


SOO 
900 
IOOO 
I IOO 
I200 
I300 
1400 
1500 
1600 
l80O 
2O0O 
24OO 


ft. 
1.56 

i-39 
1.25 
1. 14 
1.04 
0.96 
0.89 
0.83 
0.78 
0.70 
0.62 
0.52 


ft. 
O.065 
O.051 
O.040 
O.033 
O.028 
O.023 
0.020 
O.OI7 
O.O14 
O.OII 
O.OO9 
O.OO7 


8 

9 
10 

11 
12 
13 
14 

15 
16 
18 
20 
24 


ft. 
O.OIO 

O.OII 

O.OI2 
0.014 
0.015 
0.016 
0.017 
0.019 
0.020 
0.022 
0.025 
0.030 


ft. 
O.055 

O.O4O 

O.028 

0.020 

O.OI3 

O.OO7 

0.002 


ft. 

0.002 
O.O06 
O.OII 

0.016 
0.022 


ft. 
0.55 
O.40 
O.28 
0.20 
O.I3 
O.07 
0.02 


ft. 

0.02 
0.06 

O.II 

0.16 
0.22 



37* 



SURVEYING. 



1 = 50'. x = 25'. 


Sag. 


Pull. 


Resultants ± 


P 
w' 


y- 


(2S - /) 

— error. 


P 

W 


Elonga- 
tion 
-f- error. 


Error in / 


Error in 1000 ft. 


- 


+ 


- 


+ 


400 
500 
600 
700 
800 
90O 
1000 
1 100 

1200 
1300 

1400 

1500 
1600 
1700 
1800 


ft. 
0.78 
0.63 
0.52 
0.45 
0.39 
o-35 
0.31 
0.28 
0.26 
0.24 
0.22 
0.21 
0.19 
0.18 
0.17 


ft. 
O.033 

0.020 

0.014 

O.OIO 

0.007 

0.006 

O.OO4 

O.OO4 

O . OO4 

O.OO3 

0.003 

0.002 

0.002 

0.002 

O.OOI 


8 
10 
12 

14 
16 
18 
20 
22 
24 
26 
28 
30 
32 

34 
36 


ft. 
0.003 

0.003 

O.OO4 

O.OO4 

1 
O.OO5 

0.006 ! 

0.006 

0.007 

0.008 

O.ooS 

0.009 

O.OO9 

O.OIO 

O.OII 

O.OII 


O.030 
0.017 
O.OIO 

0.006 
0.002 




O.60 

0.34 
0.2I 
O.II 
O.04 






0.002 
0.003 
0.004 
0.005 
0.006 
0.007 
0.008 
0.009 

O.OIO 




O.O3 
O.06 
O.08 
O.IO 

0.12 
0.14 
0.16 
0.18 
0.20 





Assuming values of — , the formulas are readily solved for 

any assumed distance between supports and the results tabu- 
lated ; seven-place logarithms are best for this work. 

The ioo' tape is chosen because it furnishes a ready means of 
calculating a table for any other length of tape by a decimal 
reduction of the errors, per iooo', in proportion to the length 

P 

desired, and tabulated with values of — reduced in the same 

w 

proportion. There are those who use the ioo' tape free-hand, 
with 16 to 20 pounds pull, and say they do the work uniformly. 



CITY SURVEYING. 



379 



In the ordinary formula for elongation, X 



-~-r , we have 
Ek 



the section k, a multiple of w. The foregoing tables are calcu- 
lated from the value w = 3.4^. The tension in the tape F 



t 














Errors for vxti feet. , 


r? variable 














0.7 






































































































































































/ 










\\ 
































o.c 










4W 




















































































\-4 
































1 








































/ 


\ 








































0.5 


\ v 










\-y 
































1 










\6 
































* 








































\* 






































0.4 


\ 




















































\cn 






























\ -1 










\ 
































\l 
















































































OtS 














































5L 










V 


























































ii 






































b p] 


iU^. 
















\* 


- 










V 


S? 






,,r 


F2H 














Q.2 
























*B 


(•0i^ 






































































































. 






O" 












































"x» 


^^^ 


<£» 


























0.1 
















"*<% 


/ 


j£>>i 










































. ^ 


w 




















'*•> 




M 


























a/te 


















\7 


for 


fti_ 






^» 
















""■ 


— — .. 


- — . 


— . 












"" 




■.a 


50 


^a/> 






















0.0 
















1 

























2.0 



1.0 



400 



600 



800 



1000 1200 



1400 



Fig. 106. 



1600 



1800 



<souu 



&00 £ 



* E is the modulus of elasticity in pounds to the square inch, and k is the 
area of the cross-section in square inches, Z being given in the same denomina- 
tion as A. 



380 SURVEYING. 



differs from the horizontal tension P, so thatP' = P secant i 

(t = inclination to the horizontal), a second difference which 

is so small that it may be neglected. Let E = 27500000 (see 

PI $.4Pl 
Chapter XIV.), hence -^r = , nearly. 

r J Ek 27500000^ J 

The same facts for 1000 feet distance are shown in 
Fig. 106. In the tables the plus and minus errors are shown 
separately for a single length of tape only, and combined 
for iooo' feet ; in the figure they are separated for the 
whole distance and the resultants of the table are the vertical 
intercepts between the curves (minus errors) and the straight 
line (plus errors). The sag for a single length of tape and cor- 

P 

responding — is shown by dotted curved lines ; these are 

plotted to a reduced vertical scale which is shown at the right 
of the sketch. 

295. The Working Tension. — In using these tables it is 
best to measure the sag until the necessary pull for the tape is 
learned. When the ends of the tape are at a known elevation 
above a level surface, a rule at the middle of the tape will 
show whether the pull is right. The fore chainman should 
learn to pull steadily, not with a jerk, as he sticks the pin. A 
more emphatic statement than the figure itself is of the 
worthlessness of an unsteady hand at the forward end of the 
tape it would be hard to make. A consciously constant 
pull, the same every time, is necessary for good work. To ob- 
serve the sag is the surveyor's means, in the field, of knowing 
that the work is being done. He soon learns to judge with 
considerable accuracy whether the proper pull is constantly 
maintained. The proper pull is determined by the tension at 
which the tape is tested ; call this p. Then, having weighed 

. pl P 

the tape, jj/- = — • Seek the plus error from elongation for this 

P 
value of — ; then find the same plus error between the curve for 

w x 



CITY SURVEYING. 38 1 



P 
that length of tape and the straight line; the corresponding — 

is right for field use. 

For example, a 50' tape weighs six ounces, and the pull, 

when tested, was five pounds; .*. — = ^ — = 666, and the 



w 



T6" 



elongation = C/.083. The curve for a 50' tape marked — 
error from sag is distant from the line marked -f- error from 

P 

pull the same amount when— - = 1233. Whence P= 1233 

X xV "5" 5° = 9i pounds, and the sag = C/.25. When a tape 
is to be suspended freely in use, the tension at the test,/, should 
not be such that the working tension jPwill be so great as to 
be impracticable; but it is also to' be noted that slight varia- 
tions of pull do not affect the result as much, when the tension 
is considerably above the normal, as the same variations would 
affect it if the tension were at or below the normal. 

296. The Effect of Wind. — A very moderate wind has a 
marked effect on the sag of the tape ; the wind-pressure on the 
surface of tape exposed increases the sag and gives it a diago- 
nal instead of a vertical direction. The exposed surface of the 
tape constantly changes, and this results in vibrations which 
make it difficult to tell where either end of the tape is. The 
effect of its action, which is a minus error, varies approximately 
as the square of the length of tape exposed. The effect of 
winding up part of the tape so as to use a shorter length is to 
increase the use of the plumb, which is also affected by the 
wind, and the result is a loss of a part or all that is gained. A 
high working tension reduces the effect of the wind. But the 
only way to eliminate this source of error is to cease from any 
piece of work when the wind is so high that it cannot be done 
as it should be done. There are estimates, topography, etc., 
which do not require a high degree of precision and which 
can be done when other work cannot. 



382 SURVEYING. 



297. The Effect of Slope. — When the tape is used with 
its ends at different elevations, if it hangs freely its lowest 
point would not be in the middle, but nearer the lower end. 
The corrections for sag and pull still apply, however, with 
inappreciable error, for all practicable cases. The normal 
tension, therefore, remains the same as for a level tape. A 
correction must now be made, however, for the grade, the 
value of which is / vers, i, where / is the distance measured 
along the slope, and i is the angle with the horizontal. The 
measured distance is always too great by this amount* 

The available means by which the tape may be kept level 
are: (1) The judgment of two field-hands. (2) On difficult 
lines, the presence of the surveyor standing at one side where 
his position has some advantages. A distant horizon often very 
sharply defines the horizontal. (3) Where streets are im- 
proved, although it may be impracticable to measure along the 
slope, the known fall per 100 feet will give the needed infor- 
mation. (4) Where none of these methods are sufficient, test 
the judgment by plumbing at different heights and correcting 
the pin if necessary. These methods will eliminate the worst 
errors ; but where it is necessary to measure lengths of five 
or ten feet, and then plumb from above the head, the uncor- 
rected remnant will be considerable, probably that due an 
inclination of two per cent on the whole length of such 
lines, with very careful work to get so near. This difference 
in the character of lines is to be taken into the account in 
balancing the survey. Note that the resultant error is always 
minus. 

298. The Temperature Correction. — The temperature of 
the tape at the time when the work is done affects the result. 
This is not the temperature in the shade that day, nor the 



* This question is fully discussed in Chapter XIV. , where the correction is 
found in terms of the difference in elevation of the two ends. — J. B. J. 



CITY SURVEYING. 383 



reading at the nearest signal station, but is the tempera- 
ture out on the line, under the conditions which exist there. 
A grass-covered slope, descending away from the sun, will 
often show at the same time as much as twenty or thirty 
degrees lower temperature than a bare hillside inclining 
toward the sun. The thermometer is needed with the work. 
If the co-efficient of expansion is not known, use 0.0000065 
for i° F. 

It is very desirable in a city-surveyor's work that he be able 
to apply his corrections at once while in the field. If he goes 
out to measure any given distance, he must be able to fix his 
starting-point and drive his stake at the finish. If the weather 
is hot or cold, he knows what it differs from the temperature 
at which his tape is tested, and applies the correction at once 
to the whole distance. He watches that the pull is right, 
that the tape is kept horizontal, that the work stops when 
the wind is too severe, and that the checks show the desired 
accuracy. 

299. Checks. — Every piece of work should be carried on 
till it checks upon other work, verifying its accuracy within 
desired limits. This method ties up every survey at both ends. 
In order to be prepared to do this expeditiously, the surveyor 
should lay out general lines which should be joined into a sys- 
tem embracing the town-site. The lines of leading streets and 
the boundary-lines of additions give most valuable information 
when made parts of such a system. This borders on the geo- 
detic idea, but it will generally be impracticable to determine 
the lengths of these lines by triangulation from a measured 
base, for the stations can very rarely be so chosen that the 
angles can be measured upon the whole length of the lines, or 
the diagonals be observed at all. Still, the angles should be 
measured upon the best base practicable. Permanent build- 
ings and existing monuments showing the lines of intersecting 
streets should be noted both for line and distance. 



3 $4 SURVEYING. 



MISCELLANEOUS PROBLEMS. 

300. The Improvement of Streets involves — (1) The 
estimation of the earthwork in the grading and shaping of the 
street. (2) The location of the improvements along the lines of 
the dedicated streets. City ordinances usually prescribe a cer- 
tain width of sidewalks and roadway for each width of street. 
(3) The location of improvements at the grade fixed by ordi- 
nance. (4) The estimation of materials furnished by contractors 
and used in the work. The position of monuments which will 
be disturbed during the progress of the work is preserved by 
witness-stakes driven beyond the limits of disturbance. When 
this precaution is neglected it results in all sorts of angles and 
offsets in the curb-lines, in cases where there is surplus or defi- 
ciency in the original survey. Take a case improved one 
block at a time, where the first block is established by record 
distance from the right, the second block by record distance 
from the left, and a third by running from this last point to 
a point established at the end of the third block by measuring 
again from the right, etc. The resulting lines of curb will not 
give a suggestion of where the street was laid out. Some sur- 
veyors are accustomed to replace from their witness-stakes the 
monuments on the new grade. Such a practice is certainly 
to be commended ; the small cost to the public treasury can 
well be borne for the public good. 

301. Permanent Bench-marks. — In order to secure accu- 
racy and uniformity in elevations throughout a city, bench- 
marks are established by running lines of levels radiating from 
the directrix, and checking the work by cross-lines at conven- 
ient intervals, these cutting the whole territory into small par- 
cels, so that a standard bench-mark will never be far from any 
work which must be done.* This work is carried on as far as 

* These various lines of levels will form a network, such as that shown in 
Chapter XIV., which should be adjusted once for all as described in that chapter, 



CITY SURVEYING. 385 



grades are established, and generally as far as the city officers 
are prepared to propose grades for adoption by ordinance. 
There is a view of what constitutes or is essential to accurate 
methods which would make every piece of work start from 
first principles, so that it may not depend in any way upon er- 
rors involved in work previously done. But work done on this 
plan does not have to be extended very far before the results 
will show plainly that there is a wide margin between the uni- 
formity attained and the accuracy attempted. 

302. The Value of an Existing Monument is based (i) on 
the fact that it corresponds in character and position to a mon- 
ument described on the recorded plat ; (2) on the custom to 
place monuments upon the completion of a survey, and on the 
supposition that this monument in question was set in pursu- 
ance of such custom, although no monuments are noted on 
the plat ; (3) on recognition by surveyors and owners of land 
affected by it ; (4) on the knowledge that it was placed by a 
competent surveyor at a time when data were accessible which 
are not now in existence. The value of the evidence which 
establishes or tends to establish the reliability of the monument 
is primarily a question for the judgment of the surveyor. His 
decision must be reviewed and defended before courts and ju- 
ries when there is a difference of opinion. 

The monument is valueless, or less valuable in all degrees, 
when there is evidence that it has been disturbed. It some- 
times happens that there is no better way to establish a corner 
than to straighten up a stone which is leaning, but has not 
been thrown entirely out of the ground. Inquiry often brings 
out the fact that a stone, after being completely out of the 
ground, has been reset either by agreement of owners adja- 



and so one elevation obtained for each bench-mark. It is common for each 
bench-mark in a city to have numerous elevations differing by several tenths of 
a foot, and all of about equal credence. — J. B. J. 
25 



386 SURVEYING. 



cent, or by the reckless individual who did the mischief, and is 
still pointed out as the stone the surveyor set. As a recog- 
nized corner such a stone has some value, i.e., it is to be sup- 
posed that it is somewhere in the right neighborhood ; but if its 
position can be verified from other points which have not been 
disturbed the work should be retraced. If the original survey 
was made in a careless way or the corner-stones were badly 
set, they may help a careful man to come to an average line 
which shall correspond with the recorded plat. Monuments 
are sometimes moved or destroyed maliciously. It is wise for 
a surveyor to test discreetly everywhere, but to be especially 
careful where there has been quarrelling about lines. 

There is a principle, recognized to some extent by the 
courts, that the existing monument is the evidence of the orig- 
inal survey, whether or not it is called for by the recorded 
plat. The custom that the surveyor making the subdivision 
and the plat for record shall set corner-stones is so far fol- 
lowed that this is generally true, cases of accident, carelessness, 
and mischief, and such cases as that mentioned below, being 
somewhat exceptional, but many times very real. It is some- 
times attempted to go a step further and affirm that the re- 
corded plat is the record of the survey. This reverses the or- 
der of events in most cases, the survey being made in order to 
mark upon the ground the chief points of a plan already fixed 
upon ; and as to all the main lines, the plat is not altered, how- 
ever carelessly the survey may be made. There are subdivisions 
where no monuments were set and where no certain evidence 
is in existence of how or where the original survey was made, 
or whether any survey was made at all, and yet there is a re- 
corded plat. A surveyor being called upon to make a survey 
of some parcel in such a subdivision, sets stones in order to se- 
cure recognition for his theory of the proper location. If he 
does his work carefully he undoubtedly does the public a ser- 
vice. Can any amount of ignorance of when or why these 



CITY SURVEYING. 387 



stones were set ever make them evidence of the original sur- 
vey? In other cases some monuments maybe in existence, 
but more would be convenient, — points are determined from 
existing monuments in accordance with the recorded plat and 
stones are set. Another surveyor may feel a little nervous 
about manufacturing this sort of evidence of the original sur- 
vey, or more likely, may think it too much trouble and a dam- 
age to the business, for the more doubt the more work for the 
surveyor, so drives his stake. Then comes the owner who, 
desiring to secure a permanent corner, digs a hole about the 
stake without taking offsets, throws it out, and sets in a stone 
— an existing monument! This is no fancy sketch, nor are 
such facts so very rare. The young man who thinks he would 
like to be a surveyor, but has no eyes nor ears for facts like 
these, had better turn his attention to some other business. 
Surveying is an art — not an exact science.* 

303. The Significance of Possession. — Possession has a 
value in reestablishing old lines where all monuments have 
disappeared. It is a species of perpetuating testimony of their 
positions. The average of a series of improvements will often 
give a very close determination of where the corner must have 
stood. The practised eye accustomed to sharply defined lines, 
every lot having very nearly its right quantity, which are cus- 
tomary where lines are well established, will notice at once the 
irregular possession, — gaps between houses, vacant spaces 
between fences and houses, too little for use, too much for 
ornament, which may be seen where lines are in doubt and 
every man expects the next surveyor to make a conflicting 
survey. Like the men of the present, most men in the past 
have preferred to be right — have made efforts to be right — 
have employed surveyors ; we can judge where these men in 



* Consult Judge Cooley's paper on the Judicial Functions of the Surveyor, 
Appendix A. 



3^8 SURVEYING. 



the past worked from by seeing where their works are. The 
legal principle has a bearing here, that " he who would sue to 
dispossess another must first show a better title." The sur- 
veyor who attempts to dispute possession must show better 
evidence than possession of the right location of the lines he 
is employed to retrace. 

304. Disturbed Corners and Inconsistent Plats. — The 
work of testing a corner that probably has been disturbed has 
many points of likeness to the work of reestablishing corners 
that have disappeared altogether. The recorded plat is in all 
cases the basis of the work. When it records the results of a 
survey it is to be presumed that the surveyor endeavored to 
do accurate work ; hence his work, if not absolutely correct, 
was probably uniform. Lines which are shown by the plat 
as straight lines are to be retraced as straight lines. Lines in- 
volve less liability to error than measurements, and are first to 
be considered. Determine as many points as possible by 
straight lines between existing monuments. Then test the 
measurements along the extreme lines and the streets which 
are the basis of the subdivision. If the measurements between 
undoubted corners agree with the plat so closely, or if they 
differ so uniformly that the presumption of accurate work 
is justified, corners that are out of line or out of proportionate 
distance have the burden of proof against them. He who 
would claim for them authority must show that they have not 
been disturbed, and that they are consistent with some ra- 
tional location. If there was no original survey, that fact is no 
excuse for careless work at a later time ; there is always some 
place to begin. The case when the recorded plat does not 
agree with itself presents more difficulties, such as the follow- 
ing: (1) The lines do not give the same points as the distance; 
(2) The distances disagree among themselves; (3) The monu- 
ments disagree with both lines and distances impartially, or 
agree with one and disagree with the other, while the general 



CITY SURVEYING. 389 



character of the work negatives the supposition that they were 
ever carefully set. The object to be sought is not to perpet- 
uate forever the blunders of the original survey, but to seek 
the most rational adjustment of all the evidence, so that all parts 
may be located with a minimum of conflict, and so that no 
one shall be able to prove your survey wrong, i.e., show a 
more reasonable location for any part. A consultation of 
surveyors before too many conflicting interests have developed 
is often advantageous. 

305. Treatment of Surplus and Deficiency. — It is gen- 
erally a simpler problem to determine in which block differences 
of measurement, whether surplus or deficiency, belong than it 
is to know what to do with them in the matter of lot-location. 
There has never been any theory invented for the treatment 
of either surplus or deficiency which is able to stand the test 
of the courts against all combinations of circumstances. A 
few suggestions with the more probable limitations are all the 
help that can be offered : every case must be investigated for 
itself. (1) A distribution of the whole front in proportion to 
the record distances meets general approval, at least in cases of 
surplus, until it comes in conflict with possession. This is just 
the time when an owner of ground wants to know what his 
rights are, and it is also the time when no surveyor can tell 
him. A compromise, or the verdict of a petit jury, which 
passes foreknowledge, are the chief alternatives. The courts 
say that he who would sue for possession must show a better 
title. An examination shows that each has a better title 
than any other to so much ground as the plat assigns to his 
lot, but that no one has a better title than any other one to 
any part of the surplus. The surveyor does wisely to take 
note of possession and make, if he can, such a location as is in 
accordance with the record, and yet not in conflict with posses- 
sion. When this is not possible, let the map and certificate of 
survey be made in such a way that they are simply a state- 



39° SURVEYING. 



ment of the facts. It is not a surveyor's business to decide 
legal questions or give judgment in ejectment. (2) Because 
a suit for surplus will not lie, it has been thought that he who 
first took possession of the surplus would be secure if he were 
only careful to take it so that every other one might have his 
ground. Trouble with this view arises because it is not possi- 
ble to locate the surplus. When one man has appropriated all 
there is in the block, and the rest but one have appropriated 
each his proportionate share, then comes the last man. The 
more surplus in the block the more he is deficient ; he wants his 
ground, and he finds it easier to sue the one man than the twenty. 
Perhaps, in order to be sure of a case, he had better sue them all. 
The cases which arise in practice take on an infinite variety of 
complications and are not usually so simple as these described. 
(3) The fact is, that the idea that a subdivision ought to 
have a little surplus is irrational. The work should be so 
close to the standard that the surveyor who retraces the lines 
would testify : " According to the best of my knowledge and 
belief, there is neither surplus nor deficiency there. In retrac- 
ing my own work, which is carefully executed, I observe as 
great discrepancies as any which I find in this subdivision, and 
I conclude that the small difference which I observed in this 
case was as likely to have been an error in my own work as to 
attach to the subdivision." (4) Deficiency would seem to be 
easier to deal with than surplus ; for when the last man has not 
his ground he has a valid claim against the original owner for a 
rebate on the purchase-price. But the burden of the difficulty 
in this case falls on the surveyor. When a man brings his 
deed and asks a survey of lot 9, while 8 and 10 are unsold and 
lots 1 to 7 are already in possession, he leaves lot 8 its ground 
and the deficiency in lot 10. Suppose it turns out that lot 10 is 
next sold, and that the surveyor reports it deficient, the seller, 
when waited on, may reply, " I have not sold more ground in the 
block than I owned ; the surveyor has made a mistake in locat- 



CITY SURVEYING. 39 1 



ing lot 9." This liability attaches to every location which is 
made before every lot, between the one located and one corner 
of the block, is sold. (5) It is practicable for the original 
owner to so write his deeds as to locate surplus or deficiency. 
By beginning all deeds at the record distance from one street 
and continuing this uniformly through the block, the differ- 
ence goes in the lot farthest from the starting-point ; or he 
may continue the process up to any line which he may choose, 
and work from the other end of the block in deeding the re- 
maining lots ; then the difference falls upon the line chosen 
and falls to the share of the lot abutting upon that line which 
is last deeded. But to approve this method is to affirm the 
practicability of absolute accuracy in work. No one can tell 
how small a difference may cause trouble. 

306. The Investigation and Interpretation of Deeds for 
the use of the land-surveyor, dealing with the harmony or 
conflict of the descriptions, is entirely a different work from 
that of the investigator of titles, which deals with the legal 
completeness of the conveyance. In the older parts of a town 
the deed of the present proprietor frequently does not give 
information sufficient to fix the correct location. The key 
may lie in some boundary in an early deed referring to a still 
earlier conveyance of adjacent property. Or the earlier deeds 
may give clearly defined locations, while the latter ones 
say " more or less" at every point. In some cases the deeds 
are in such a condition that it is impossible to tell what they 
mean until it is known what the possession is. Skill in this 
work can only come after considerable experience ; local prac- 
tices must largely determine what is necessary. 

307. Office Records. — The surveyor's office when well 
planned is so arranged that no item of information which 
promises to be useful shall be lost. The customary methods 
of indexing, and of block-plats for keeping notes, do not take 
a very firm hold on general lines or the connections between 



39 2 SURVEYING. 



subdivisions ; they fail, in fact, in that part of the work which 
has the most vital relation to efforts at future improvement. 
It is advisable to add to the block-plats and indexes a general 
atlas of the whole town for office use, at a scale of say ioo' 
to the inch, so that an area nearly half a mile square may 
appear on the open pages. Such an atlas may show the notes 
of the general lines and their angles, the base-line measure- 
ments, the relation of subdivisions to one another, and a 
variety of other information which it is difficult to pick out in 
the widely scattered field-notes which first gathered the in- 
formation, and which, with their larger scale, the block-plats 
are not well adapted to show in a connected form. 

There are filed in connection with deeds many plats which 
do not appear on the record plat-books of the recorder's 
office ; these need to be indexed, or, better, abstracted for 
office use. 

The field-notes, when prepared for the surveyor's use in 
the field, should show in an accessible and portable form all 
the information which the office contains and which is rele- 
vant to the survey in hand. Labor spent beforehand in a 
thorough preparation of accessible information is labor saved. 

308. The Preservation of Lines after the monuments 
have disappeared is accomplished by means of notes on build- 
ings, marks and notes on curbing, paving, fences, etc. Notes 
on buildings describe not only the character of the building, 
but the particular part noted, so that another man, years 
afterward, using the same note would have no doubt of the 
identity of the part. In a growing town the work of keeping 
up the notes goes on without ceasing, — buildings are remod- 
elled or rebuilt, streets reconstructed, destroying old marks. 
The old becomes the new so constantly that the surveyor 
who would preserve the information which he already has 
must be constantly employed at the work of renewal. There 
is no place either in the street or out of it where the surveyor 



CITY SURVEYING. 393 



can place his mark and say to all comers, " Touch not." It 
follows that whenever it is necessary to use any mark, about 
the permanence of which there can be a shadow of a doubt, 
the permanence of the mark must be shown by some prac- 
ticable test ; it is careless to assume it. 

309. The Want of Agreement between Surveyors arises 
from differences of information or of judgment, and in a less 
degree from differences of skill. These are all just as human 
elements as the lawyer deals with in his work. Testimony is 
affected by the interests of those who speak, and the judg- 
ment varies with the temperament of the individual. Per- 
haps one of the most difficult lines for a surveyor to draw is 
that which separates his confidence in his own skill in retrac- 
ing a survey which was confessedly inaccurate, from his re- 
liance on testimony which is evidently biassed as to the posi- 
tion or disturbance of monuments, and other facts which 
may help him to form a correct judgment. Errors in execu- 
tion may be kept within such limits that work which shows 
differences in closing of 1 in 5000 should be retraced, and the 
average observed differences in one surveying party's work 
will not exceed I in 20000. Two sets of men working to 
reach the same standard may err in opposite directions, so 
that differences between two surveyors may reasonably be 
expected to be somewhat larger than either would tolerate in 
his own work. 



CHAPTER XIII. 
THE MEASUREMENT OF VOLUMES. 

310. Proposition. — The volume of any doubly-truncated 
prism or cylinder, bounded by plane ends, is equal to the area of a 
right section into the length of the element through the centres of 
gravity of the bases, or it is equal to the area of either base into 
the altitude of the element joining the centres of gravity of the 
bases \ measured perpendicular to that base. 

Let ABCD, Fig. 107, be a cylinder, cut by the planes OC 
and OB y the unsymmetrical right section EF being shown in 
plan in E ' F '. Whatever position the cutting planes may have, 
if they are not parallel they will intersect in a line. This line 
of intersection may be taken perpendicular to the paper, and 
the body would then appear as shown in the figure, the line 
of intersection of the cutting planes being projected at O. 

Let A = area of the right section ; 

A A = any very small portion of this area ; 
x = distance of any element from O ; 
then ax = height of any element at a distance x from O. 



An elementary volume would then be axAA, and the total 
volume of the solid would be 2axdA. 

Again, the total volume is equal to the mean or average 
height of all the elementary volumes multiplied by the area 
of the right section. 

The mean height of the elementary volumes is, therefore, 



THE MEASUREMENT OF VOLUMES. 



395 



*2axAA a^xAA 



But 



2xAA 



is the distance from to the 



A ~ A ' ~"" ^ 
centre of gravity, £, of the right section,* and a times this dis- 
tance is the height of the element LK through this point. 
Therefore, the mean height is the height through the centre of 




\a 



O *a=r;rj7 



-x- 



E JG 




Fig. 107. 

gravity of the base, and this into the area of the right section 
is the volume of the truncated prism or cylinder. The truth 
of the alternative proposition can now readily be shown. 

Corollary. When the cylinder or prism has a symmetrical 
cross-section, the centre of gravity of the base is at the centre 
of the figure, and the length of the line joining these centres 
is the mean of any number of symmetrically chosen exterior 
elements. For instance, if the right section of the prism be a 
regular polygon, the height of the centre element is the mean 
of the length of all the edges. This also holds true for paral- 
lelograms, and hence for rectangles. Here the centres of gravity 

* This is shown in mechanics, and the student may have to take it for 
granted temporarily. 



39 6 



SURVEYING. 



of the bases lie at the intersections of the diagonals ; and since 
these bisect each other, the length of the line joining the in- 
tersections is the mean of the lengths of the four edges. The 
same is true of triangular cross-sections. 

311. Grading over Extended Surfaces. — Lay out the 
area in equal rectangles of such a size that the surfaces of the 
several rectangles may be considered planes. For common 
rolling ground these rectangles should not be over fifty feet 
on a side. Let Fig. 108 represent such an area. Drive pegs at 



4 


4 


4 


4 


4 




4 


4 


4 








4 


4 


4 




3 
2 


2 1 













2 

Fig. 108. 



the corners, and find the elevation of the ground at each in- 
tersection by means of a level, reading to the nearest tenth of 
a foot, and referring the elevations to some datum-plane below 
the surface after it is graded. When the grading is completed, 
relocate the intersections from witness-points that were placed 
outside the limits of grading, and again find the elevations at 
these points. The several differences are the depths of excava- 
tion (or fill) at the corresponding corners. The contents of 
any partial volume is the mean of the four corner heights into 
the area of its cross-section. But since the rectangular areas 
were made equal, and since each corner height will be used as 
many times as there are rectangles joining at that corner, we 
have, in cubic yards, 



V = 



A 



AX 27 



[^ + 2^ + 3^ + 4^]. • • (1) 



THE MEASUREMENT OF VOLUMES. 



397 



The subscripts denote the number of adjoining rectangles 
the area of each of which is A. 

From this equation we may frame a 

RULE. — Take each corner height as many times as there 
are partial areas adjoining it, add them all together, and divide 
by one fourth of the area of a single rectangle. This gives the 
volume in cubic feet. To obtain it in cubic yards, divide by 
twenty-seven. 

If the ground be laid out in rectangles, 30 feet by 36 feet, 

A 1080 , ._ , , 

then——- — = — x- =10; and if the elevations be taken to 
4 X 27 108 

the nearest tenth of a foot, then the sum of the multiplied 

corner heights, with the decimal point omitted, is at once the 

the amount of earthwork in cubic yards. This is a common 

way of doing this work. In borrow-pits, for which this method 

is peculiarly fitted, the elementary areas would usually be 

smaller. 

In general, on rolling ground, a plane cannot be passed 

through the four corner heights. We may, however, pass a 

plane through any three points, and so with four given points 




on a surface either diagonal may be drawn, which with the 
bounding lines makes two surfaces. If the ground is quite 
irregular, or if the rectangles are taken pretty large, the sur- 
veyor may note on the ground which diagonal would most 



398 



SUE VE YING. 



nearly fit the surface. Let these be sketched in as shown in 
Fig. 109. Each rectangular area then becomes two triangles, 
and when computed as triangular prisms, each corner height 
at the end of a diagonal is used twice, while the two other 
corner heights are used but once. That is, twice as much 
weight is given to the corner heights on the diagonals as to 
the others. In Fig. 109, the same area as that in Fig. 108 is 

^ h 2 shown with the diagonals drawn which best fit 

the surface of the ground. The numbers at 
the corners indicate how many times eacl 
height is to be used. It will be seen that 
k* each height is used as many times as there are 
triangles meeting at that corner. To derive 
the formula for this case, take a single rectangle, as in Fig. 
no, with the diagonal joining corners 2 and 4. Let A be th< 
area of the rectangle. Then from the corollary, p. 395, w< 
have for the volume of the rectangular prism, in cubic yards, 




Fig. no. 



V 



A f kt + /l2 + /li k^ + ks + k. 



2 X 27 

A 
6x27 



3 ' 3 

{K + *K + k 3 + 2k 4 ). ...... (2; 



For an assemblage of such rectangular prisms as shown in 
Fig. 109, the diagonals being drawn, we have, in cubic yards, 



V = 



6 X 27 



\2k x + 22h % + 3^3 + \2K + 5Zk 6 

+ 62k 6 + 72k, + S2k 6 -}; ... (3) 



where A is the area of one rectangle, and the subscripts denote 
the number of triangles meeting at a corner. 



THE MEASUREMENT OF VOLUMES. 399 

As a check on the numbering of the corners, Fig. 109, add 
them all together and divide by six. The result should be 
the number of rectangles in the figure. In this case, if the 
rectangles be taken 36 feet by 45 feet, or, better, 40 feet by 40.5 
feet, then the sum of the multiplied heights with the decimal 
point omitted is the number of cubic yards of earthwork, the 
corner heights having been taken out to tenths of a foot. 

The method by diagonals is more accurate than that by 
rectangles simply, the dimensions being the same ; or, for 
equal degrees of exactness larger rectangles may be used with 
diagonals than without them, and hence the work materially 
reduced. In any case some degree of approximation is neces- 
sary. 

312. Approximate Estimates by means of Contours. — 
(A) Whenever an extended surface of irregular outline is to 
be graded down, or filled up to a given plane (not a warped or 
curved surface), a near approximation to the amount of cut or 
fill may be made from the contour lines. In Fig. 11 1 the full 
curved lines are contours, showing the original surface of the 
ground. Every fifth one is numbered, and these were the con- 
tours shown on the original plat. Intermediate contours one 
foot apart have been interpolated for the purpose of making 
this estimate. The figures around the outside of the bound- 
ing lines give the elevations of those points after it is graded 
down. The straight lines join points of equal elevation after 
grading ; and since this surface is to be a plane these lines are 
surface or contour lines after grading. Wherever these two 
sets of contour lines intersect, the difference of their elevations 
is the depth of cut or fill at that point. If now we join the 
points of equal cut or fill (in this case it is all in cut), we ob- 
tain a new set of curves, shown in the figure by dotted lines, 
which may be used for estimating the amount of earthwork. 
The dotted boundaries are the traces on the natural surface of 
planes parallel to the final graded surface which are uniformly 



4oo 



SURVEYING. 



spaced one foot apart. These areas are measured by the 
planimeter and called^, A if A^ etc. Each area is bounded by 
the dotted line and the bounding lines of the figure, since on 
these bounding lines all the projections of all the traces unite, 




the slope here being vertical. For any two adjoining layers 
we have, by the prismoidal formula* as well as by Simpson's 
one-third rule, 



V I . 3 =^(A l + 4 A, + A t ), 



(i) 



where h is the common perpendicular distance between the 
sections. 



* For the demonstration of the prismoidal formula see p. 403. 



THE MEASUREMENT OF VOLUMES. 4OI 



For the next two layers we would have, similarly, 

n-5 = I (A,+4A t + A t ); ; (2) 

or for any even number of layers we would have, in cubic 
yards, 

V = J^7 {Ai + 4A ' + 2A >+4A t + 2A t + A„), (3) 

where n is an odd number, h and A being in feet and square 
feet respectively. 

{E) Whenever the final surface is not to be a plane but a 
surface which may be defined by drawing the contours lines as 
they are to be when the grading is completed, the above 
method may still be used. Thus, suppose a given tract of 
ground, the contours of which have been carefully determined, 
is to be transformed into certain new outlines, as is often 
required in landscape-gardening and in the making of parks 
and cemeteries, the new contours may be traced on the plat 
containing the original contours by using a different-colored 
ink. The second set of contours are now curved instead of 
straight, as was the case in the preceding example. Otherwise 
there is no difference in the methods. The intersections of the 
two sets of contours are marked with the number of feet of 
cut or fill, the same as before, the cuts being designated by a 
plus and the fills by a minus sign. The curves of equal cut or 
fill are now drawn, preferably in an ink of a different color from 
the other two, and areas measured and the volume computed 
exactly as in the former case. It would also be well to desig- 
nate the cut and the fill curves by ink of different shades but 
of the same color. 

When a rectangular area, as a city block, is to be graded to 

26 



402 SURVEYING. 



a warped surface, which it generally is, the contours of this 
surface are readily obtained from the street-grades, and the 
above method used. For accurate measurements, such as 
should be made the basis of payment, the area should be di- 
vided into rectangles, as previously described. These* approxi- 
mate methods serve well for preliminary estimates. They 
may be found useful in determining street-grades when it is 
desired to equalize the cuts and fills over the blocks rather 
than on the street-lines. 

313. The Prismoid is a solid having parallel end areas, 
and may be composed of any combination of prisms, cylinders, 
wedges, pyramids, or cones or frustums of the same, whose 
bases and apices lie in the end areas. It may otherwise be 
defined as a volume generated by a right-line generatrix mov- 
ing on the bounding lines of "two closed figures of any shapes 
which lie in parallel planes as directrices, the generatrix not 
necessarily moving parallel to a plane director. Such a solid 
would usually be bounded by a warped surface, but it can 
always be subdivided into one or more of the simple solids 
named above. 

Inasmuch as cylinders and cones are but special forms of 
prisms and pyramids, and warped surface solids may be divided 
into elementary forms of them, and since frustums may also 
be subdivided into the elementary forms, it is sufficient to say 
that all prismoids may be decomposed into prisms, wedges, 
and pyramids. If a formula can be found which is equally 
applicable to all of these forms, then it will apply to any com- 
bination of them. Such a formula is called 

314. The Prismoidal Formula. 

Let A = area of the base of a prism, wedge, or pyramid ; 
A x A m , At = the end and middle areas of a prismoid, or of any 
of its elementary solids ; 
h = altitude of the prismoid or elementary solid. 



THE MEASUREMENT OF VOLUMES. 403 



Then we have, 
For Prisms, 



For Wedges, 



For Pyramids, 



V=M=^(A 1 + 4 A m + A i ) (1) 



V= k f = ^(A 1 + 4 A m + A i ) (2) 



hA k 

V= T = ? (A J + 4 A m + A J ) (3) 



Whence for any combination of these, having all the common 
altitude h, we have 

V=^A 1 + 4 A m + A,) (4) 

which is the prismoidal formula. 

It will be noted that this is a rigid formula for all prismoids. 
The only approximation involved in its use is in the assump- 
tion that the given solid may be generated by a right line 
moving over the boundaries of the end areas. 

This formula is used for computing earthwork in cuts and 
fills for railroads, streets, highways, canals, ditches, trenches, 
levees, etc. In all such cases, the shape of the figure above 
the natural surface in the case of a fill, or below the natural 
surface in the case of a cut, is previously fixed upon, and to 
complete the closed figure of the several cross-section areas 
only the outline of the natural surface of the ground at the 
section remains to be found. These sections should be located 
so near together that the intervening solid may fairly be as- 



404 SURVEYING. 



sumed to be a prismoid. They are usually spaced ioo feet 
apart, and then intermediate sections taken if the irregularities 
seem to require it. 

The area of the middle section is never the mean of the 
two end areas if the prismoid contains any pyramids or cones 
among its elementary forms. When the three sections are 
similar in form, the dimensions of the middle area are always 
the means of the corresponding end dimensions. This fact 
often enables the dimensions, and hence the area of the middle 
section, to be computed from the end areas. Where this can- 
not be done, the middle section must be measured on the 
ground, or else each alternate section, where they are equally 
spaced, is taken as a middle section, and the length of the 
prismoid taken as twice the distance between cross-sections. 
For a continuous line of earthwork, we would then have, in 
cubic yards, 



V= ^(A 1 + 4 A,+2A,+ 4 A i +2A t + 4 A s . . +A n ), . (i) 



where / is the distance between sections in feet. This is the 
same as equation (3), p. 401. Here the assumption is made 
that the volume lying between alternate sections conforms 
sufficiently near to the prismoidal forms. 

315. Areas of Cross-sections. — In most cases, in practice 
at least, three sides of a cross-section are fixed by the conditions 
of the problem. These are the side slopes in both cuts and 
fills, the bottom in cuts and the top in embankments, or fills. 
It then remains simply to find where the side slopes will cut 
the natural surface, and also the form of the surface line on the 
given section. Inasmuch as stakes are usually set at the points 
where the side slopes cut the surface, whether in cut or fill, 
such stakes are called slope-stakes, and they are set at the time 






THE MEASUREMENT OF VOLUMES. 4O5 

the cross-section is taken. The side slopes are defined as so 
much horizontal to one vertical. Thus a slope of \\ to 1 means 
that the horizontal component of a given portion of a slope- 
line is ii times its vertical component, the horizontal com- 
ponent always being named first. The slope-ratio is the ratio 
of the horizontal to the vertical component, and is therefore 
always the same as the first number in the slope-definition. 
Thus for a slope of 1 \ to 1 the slope-ratio is if. 

316. The Centre and Side Heights. — The centre heights 
are found from the profile of the surface along the centre line, 
on which has been drawn the grade line of the proposed work. 
These are carefully drawn on cross-section paper, when the 
height of grade at each station above or below the surface line 
can be taken off. These centre heights, together with the 
width of base and side slopes in cuts and in fills, are the neces- 
sary data for fixing the position of the slope-stakes. When 
these are set for any section as many points on the surface 
line joining them maybe taken as desired. In ordinary rolling 
ground usually no intermediate points are taken, the centre 
point being already determined. In this case three points in 
the surface line are known, both as to their distance out from 
the centre line and as to their height above the grade line. 
Such sections are called " three-level sections," the surface lines 
being assumed straight from the slope-stakes to the centre 
stake. 

317. The Area of a Three-level Section. 
Let d and d' be the distances out, and 

h and h' the heights above grade of right and left slope- 
stakes, respectively ; 
D the sum of d and d\ 
c the centre height, 
r the slope-ratio, 
w the width of bed. 



406 



SURVEYING. 



Then the area ABCDE is equal to the sum of the tour trian- 
gles AEw, BCw, wCD, and wED. Or, 



A = 



(d-\-d')c+(k + k') 



w 



(0 



This area is also equal to the sum of the triangles FCD and 
FED, minus the triangle AFB. Or, 



A=[c+- 



w\ D 



2TJ 2 



W 



(*) 




Fig. 



Equation (2) can also be obtained directly from equation 

(1) by substituting for h and h! in (1) their values in terms of 

. w 
a 

d and w, h = , and then putting D = d-\- d r . Equation 

(2) has but two variables, c and D, and is the most convenient 
one to use. 

318. Cross-sectioning. — It will be seen from Fig. 1 12 that 
in the case of a three-level section the only quantities to be 
determined in the field are the heights, h and h f > and the dis- 
tances out, d and d f , of the slope-stakes. These are found by 
trial. A levelling instrument is set up so as to read on the 



. THE MEASUREMENT OF VOLUMES. 407 

three points C, D, E, and the rod held first at D. The reading 
here gives the height of instrument above this point. Add 
this algebraically to the centre height (which may be negative, 
and which has been obtained from the profile for each station), 
and the sum is the height of instrument above (or below). the 
grade line. If the ground were level transversely, the distance 
out to the slope-stakes would be 

w 
d = cr A — . 

1 2 

But this is not usually the case, and hence the distance out 
must be found by trial. If the ground slopes ■! w ( 

from the centre line in a \ , \ the distance out will evidently 



v cut f 

be more than that given by the above equation, and vice versa. 
The rodman estimates this distance, and holds his rod at a cer- 
tain measured distance out, d x . The observer reads the rod, 
and deducts the reading from the height of instrument above 
grade (or adds it to the depth of instrument below grade), and 
this gives the height of that point, h x> above or below grade. Its 

1 w 

distance out, then, should be d — h x r -] — . If this be more than 

the actual distance out, d lt the rod is set farther in ; if less, it is 
moved out. The whole operation is a very simple one in prac- 
tice, and the rodman soon becomes very expert in estimating 
nearly the proper position the first time. 

In heavy work — that is, for large cuts or fills, and for irregu- 
lar ground — it may be necessary to take the elevation and dis- 
tance out of other points on the section in order to better 
determine its area. These are taken by simply reading on the 
rod at the critical points in the outline, and measuring the dis- 
tances out from the centre. The points can then be plotted 



408 SURVEYING. 



on cross-section paper and joined by straight or by free-hand 
curved lines. In the latter case the area should be deter- 
mined by planimeter. 

319. Three-level Sections, the Upper Surface con- 
sisting of two Warped Surfaces. — If the three longitudinal 
lines joining the centre and side heights on two adjacent three- 
level sections be used as directrices, and two generatrices, one 
on each side the centre, be moved parallel to the end areas as 
plane directers, two warped surfaces are generated, every cross- 
section of which parallel to the end areas is a three-level sec- 
tion. These same surfaces could be generated by two longi- 
tudinal generatrices, moving over the surface end-area lines as 
directrices. In this case the generating lines would not move 
parallel to a plane directer, but each would move so as to cut 
its directrices proportionally. The surface would therefore be 
a prismoid, and its exact volume would be given by the pris- 
moidal formula. The middle area in this case is readily found, 
since the centre and side heights are the means of the corre- 
sponding end dimensions. 

The prismoidal formula, 

V = 6xlj^ + 4Am + A ^ •■•'.". (I) 

could therefore be written 

w\ 



V= 



12 X27 



■+s^+(*+g* 



2rj 



l<u? 



1 / 1 w \ t^ L ™ r \ 



4 X 2jr 



This equation is derived directly from eq. (1) above, and eq. 

w 
(2), p. 406. The quantity — is the distance from the grade-plane 



THE MEASUREMENT OF VOLUMES. 409 

to the intersection of the side slopes, and is a constant for any- 
given piece of road. It would have different values, however, 
in cuts and fills on the same line. 
For brevity, let 

« =e% . an d -^ = l ^= K . 
2r 4 X 2jr 54 

Here K is the volume of the prism of earth, 100 feet long, in- 
cluded between the roadbed and side slopes. It is first in- 
cluded in the computation and then deducted. It is also a 
constant for a given piece of road. 
Equation (2) now becomes 

V =T^j^+^ D M^+c,)D^ A {c m +c a )D m -\-K, . (3) 

where c m and D m are the means of c x c 2 and D t D 2 , respectively. 
This equation involves but two kinds of variables, c and D, 
and is well adapted to arithmetical, tabular, or graphical com- 
putation. Thus if / = 100 ; w = 18 ; and r = 1^ ; then c = 6 ; 
and K= 200; and equation (3) becomes 

y-=m k*. + 6 )^. + fo + 6 )A + 4fm + 9 aj - 200 . ( 4 ) 

If the total centre heights (to intersection of side slopes) be 
represented by C v C 2 , and C m , then eq. (3) becomes, in general, 

V=X'(C 1 V 1 + CJD. + 4C m D m ) -K.. . . (5) 

where K' = -J^-J , and is independent of width of bed and of 
slopes. 

For any given piece of road, the constants K, K\ and C are 
known, and for each prismoid the £7's and D's are observed, 
hence for any prismoid all the quantities in eq. (5) are known. 



4io 



SUR VE YING. 



320. Construction of Tables for Prismoidal Computa- 
tion. — If a table were prepared giving the products K CD for 
various values of C and D, it 'could be used for evaluating 
equation (3), which is the same as equation (5). The argu- 
ments would be the total widths (.Z?,), and the centre heights 
(Cj). Such a table would have to be entered three times for 
each prismoid, first with C x and D 1 ; second with (7 2 and D 2 ; 
and finally with C m and D m . If four times the last tabular 
value be added to the sum of the other two, and K subtracted, 
the result is the true volume of the prismoid. 

VALUES OF Co (= -) AND K (= h °* ) FOR VARIOUS WIDTHS 
\ 2rJ \ 4 X 277-/ 

AND SLOPES. 



Width 

of 
Road- 
bed. 


Slopes. 


C 
20 


to 1. 

K 

37o 


^ tol 


% to 1. 


1 to 


1. 


\ X A tol. 


1H tol. 


lfctol. 


2 to 1. 


Co 
10 


K 

185 


C 

6. 7 


K 
123 


Co 


K 


Co 


K 

74 


Co 

3-3 


K 
62 


Co 

2.9 


K 

53 


Co 
2-5 


K 
46 


10 


5 


93 


4.0 


11 


22 


448 


11 


224 


7-3 


149 


5-5 


112 


4.4 


90 


3-7 


75 


3-i 


64 


2.8 


56 


12 


24 


533 


12 


266 


8.0 


178 


6.0 


*33 


4.8 


107 


4.0 


89 


3-4 


76 


3-o 


67 


13 


26 


626 


13 


3*3 


8-7 


209 


6-5 


*57 


5-2 


125 


4-3 


104 


3-7 


89 


3-2 


78 


14 


28 


725 


14 


363 


9-3 


242 


7.0 


181 


5-6 


145 


4-7 


121 


4.0 


104 


3-5 


91 


15 


3° 


833 


15 


4i7 


10. 


278 


7-5 


208 


6.0 


167 


5-o 


*39 


4-3 


119 


3-8 


104 


16 


3 2 


948 


16 


474 


10.7 


316 


8.0 


237 


6.4 


190 


5-3 


158 


4.6 


135 


4.0 


118 


17 


34 


1070 


17 


535 


"•3 


357 


8.5 


268 


6.8 


214 


5-7 


178 


4-9 


r 53 


4.2 


i34 


18 


36 


1200 


18 


600 


12.0 


400 


9.0 


300 


7.2 


240 


6.0 


200 


5-i 


171 


4-5 


1 5° 


19 


38 


J 337 


19 


668 


12.7 


446 


9-5 


334 


7.6 


267 


6-3 


223 


4-4 


191 


4.8 


167 


30 


40 


1481 


20 


740 


J 3-3 


494 


10. 


370 


8.0 


296 


6.7 


247 


5-7 


212 


5-o 


185 


21 


42 


^33 


21 


816 


14.0 


544 


10.5 


408 


8.4 


327 


7.0 


272 


6.0 


233 


5-2 


204 


22 


44 


*793 


22 


896 


14.7 


598 


11. 


448 


8.8 


359 


7-3 


299 


6.3 


256 


5-5 


224 


23 


46 


J 959 


23 


980 


15-3 


653 


"•5 


490 


9.2 


392 


11 


326 


6.6 


280 


5-8 


245 


24 


48 


2i34 


24 


1067 


16.0 


711 


12.0 


534 


9.6 


427 


8.0 


356 


6.9 


3°5 


6.0 


267 


25 


50 


2315 


25 


1158 


16.7 


772 


12.5 


579 


10. 


463 


8-3 


386 


I- 1 


33 1 


6.2 


264 


26 


52 


2504 


26 


1252 


i7-3 


835 


13.0 


626 


10.4 


501 


8.7 


417 


7-4 


358 


6-5 


3!3 


27 


54 


2700 


27 


1350 


18.0 


900 


r 3-5 


675 


10.8. 


54° 


9.0 


450 


7-7 


386 


6.8 


338 


28 


56 


2904 


28 


1452 


18.7 


968 


14.0 


726 


11. 2 


581 


9 3 


484 


8.0 


4i5 


7.0 


363 


29 


58 


3"5 


29 


1558 


J 9-3 


1038 


14-5 


779 


11. 6 


623 


9-7 


5i9 


8-3 


445 


7.2 


389 


30 


60 


3333 


30 


1667 


20.0 


IIII 


15.0 


833 


12.0 


667 


10. 


556 


8.6 


476 


7-5 


4i7 



THE MEASUREMENT OF VOLUMES. 



4H 



Table X.* is such a table, computed for total centre heights 
from 1 to 50 feet, and for total widths from 1 to 100 feet. 
In railroad work neither of these quantities can be as small as 
one foot, but the table is designed for use in all cases where 
the parallel end areas may be subdivided into an equal number 
of triangles or quadrilaterals. 

Example I. Three- level Ground having two Warped Surfaces. — Find the 
volume of two prismoids of which the following are the field-notes, the width 
of bed being 20 feet, and the slopes \\ to 1. 



Station 11. 
Station 12. 
Station 12 -\- 56. 



28. 9 f 



43-0 



+ 12.6 


+ 18.6 


-j- 22.0 


27.1 





40.3 


+ 11. 4 


+ 14.8 


-f- 20.2 


24-3 





34-9 



+ 9-5 



+ 10.3 +16.6 



From the table, p. 410, giving values of Co and LC, we find for w = 20, 
and r = i$, C = 6.7, and K '= 247. 

The computation may be tabulated as follows: 



Sta. 


Width, 
D=d+d'. 


Height, 
C — c-\-c . 


Partial Volume. 


Volume of 
Prismoid. 


II 


71.9 


25-3 


502 






M 


69.6 


23-4 


503 X 4 = 2012 






12 


67.4 


21.5 


447 












3021 — 


247 


2744 


M 


63-3 


19.2 


374 X 4 = 1496 






12 + 56 


59-2 


17.0 


311 






> 






.56(2254 - 


247) 


1 124 

1 



* Modeled somewhat after Crandall's Tables, but adapted to give volumes 
by the Prismoidal Formula at once instead of by the method of mean end areas 
first and correcting by the aid of another table to give prismoidal volumes, as 
Prof. Crandall has done. 

f The numerators are the distances out, and the denominators are the heights 
above grade, + denoting cut and — fill. 



412 



SURVEYING. 



Entering the table (No. X.) for a width of 71 and a height of 25, we find 
548, |o which add 7 for the 3 tenths of height, and 7 more for the 9 tenths in 
width, both mentally, thus giving 562 cu. yds. for this partial volume. Simi- 
larly for the width 67.4, and height 21.5, obtaining 447 cu. yds. The correspond- 
ing result for the middle area is 503, which is to be multiplied by 4, thus giving 
2012 cu. yds. The sum of these is 3021 cu. yds., from which is to be subtracted 
the constant volume K, which in this case is 247 cu. yds., leaving 2774 cu. yds. 
as the volume of the prismoid. 

The next prismoid is but 56 feet long, but it is taken out just the same as 
though it were full, and then 56 hundredths of the resulting volume taken. 
The data for the 12th station is used in getting this result without writing it 
again on the page. 

Example 2. Five-level Grottnd having four Warped Surfaces. — Find the 
volume of a prismoid of which the following are the field-notes, the width of 
bed being 20 feet, and the slopes i£ to 1: 



11. 



12. 



28.9 

+ I2!6 

27.1 



150 
-j- 12.0 

12.5 



+ 18.6 



20.0 
-J- 21.0 

18.5 



43-Q 
-f- 22.0 

40-3 



-L-11.4 + 12.0 +148 +19.6 -j -20 - 2 

This is the same problem as the preceding, with intermediate heights added. 
To compute this from the table, it is separated into three prismoids, as shown 
in Fig. 113. 




Let ABDGCFE be the cross-section. This may be separated into the tri- 
angle ABC, and the two quadrilaterals BCGD and ACFE. These areas are 
one half the areas of their enclosing rectangles, which areas are respectively 
cw, kd, and k'd'. Therefore the area of the section is 

A = i(cw + &d+k'd') (1) 

If the adjacent section is also a five-level section, its area maybe represented 
by an equation similar to this, and the included prismoid may be computed in 
three parts, corresponding to the three pairs of partial end areas. In this case 
the prism included between the road-bed and side slopes, whose volume is K, 
is not included, and hence it is not deducted from the result. The computation 
by Table X. would therefore be, in tabular form — 



THE MEASUREMENT OF VOLUMES. 



413 



Sta. 


d 1 . 
28.9 


k 1 . 


c. 


k. 


d. 


Partial Volumes. 


Total 

Volume. 


II 


I2.0 


18.6 


21.0 


43-o 


108 -}- 115 -f- 279 = 


502 




M 


28. Q 


I2.0 


16,7 


20.3 


41.6 


4 (104 -|- io2 + 2 6o) = 


1864 




12 


27.I 


I2.0 


14. s 


19.6 


40.3 


IOO + 91 -f 243 =e 


434 


2800 



The use of the table is the same as before, the arguments now being k' and 
d' ; c and w ; and k and */. The dimensions of the mid-area are the means of 
the corresponding end dimensions, as before. If one end area is a three-level 
section, and the next one a five-level, it is computed as a five-level section, 
only the position of the intermediate heights on the three-level section must be 
noted on the ground, these being the apices of the pyramids whose bases lie in 
the five-level section. Partial stations are computed first as whole stations, 
and then multiplied by their length and divided by 100 as before. 

By the aid of the table five-level sections are computed with great rapidity 
and accuracy. 

321. Three-level Sections, the Surface divided into 
four Planes by Diagonals. — If the surface included between 
two three-level sections be assumed to be made up of four 
planes formed by joining the centre height at one end with a 
side height at the other end section 
on each side the centre line (Fig. 1 14), 
these lines being called diagonals, 
an exact computation of the volume 
is readily made without computing 
the mid-area. Two diagonals are 
possible on each side the centre line 
but the one is drawn which is ob- 
served to most nearly fit the sur- 
face. They are noted in the field 
when the cross-sections are taken. 




Fig. 



114. 



The total volume of such a prismoid in cubic * yards is 



V = 



6x^ 7 L W + *>* + ^ + *>* + DC + UC 



w 



+ =01, +A. + IT+ K + K + H') 



->:- 



(1) 



* For % demonstration of this formula see Henck's Field-Book. 



414 



SURVEYING. 



where c l9 h„ and h x ' are the centre and side heights at one sec- 
tion and d x and df the distances out, c 2) k 2 ', h„ d v and d. 2 ' be- 
ing the corresponding values for the other end section. C and 
C are the centre heights, H and H' the side heights, and D 
and D' the distances out on the right and left diagonals. 
Although this formula seems long, the computations by it are 
very simple. Thus let the volume be found from the following 
field-notes for a base of 20 feet and side slopes \\ to 1. 



47-5 




A n 



The upper figures indicate the distances out and those 
below the lines the heights, the plus sign being used for cuts. 
The computation in tabular form is as follows : 



Sta. 


d. 


A. 


c. 


h'. 


d'. 


d+d'. 


(d + d')c. 


DC. 


D'C. 


I 


22 


8 


8 


25 


47-5 


69.5 


556 


• • ♦ • 


.... 


2 


34 


16 
H 


4 
+ /** 


4 

= 24 
= 12 


16 


50.0 


200 

88 
128 


88 


128 






-2A's = 65 X 10 
2 


c 

1 


= 650 

) 162200 
57 ) 27033 


















IOOI 


cu. yarc 


s. 



The great advantage of the method consists in the datai 
all being at hand in the field-notes. 

Hudson's Tables * give volumes for this kind of prismoid. 



* Tables for Computing the Cubic Contents of Excavations and Embank- 
ments. By John R. Hudson, C.E. John Wiley & Sons, New York, 1884. 



THE MEASUREMENT OF VOLUMES. 415 



They furnish a very ready method of computing volumes when 
this system is used. 

322. Comparison of Methods by Diagonals and by 
Warped Surfaces. — Although the surveyor has a choice of 
two sets of diagonals when this method is used, the real surface 
would usually correspond much nearer the mean of the two pairs 
of plane surfaces than to either one of them. That is, the 
natural surface is curved and not angular, and therefore it is 
probable that two warped surfaces joining two three-level sec- 
tions would generally fit the ground better than four planes, 
notwithstanding the choice that is allowed in the fitting of the 
planes. More especially must this be granted when the truth 
of the following proposition is established. 

Proposition : The volume included between two three-level 
sections liaving their corresponding surface lines joined by 
warped surfaces, is exactly a mean between the two volumes 
formed between the same end sections by the two sets of planes re- 
sulting from the two sets of diagonals which may be drawn. 

If the two sets of diagonals be drawn on each side the 
centre line and a cross-section be taken parallel to the end 
areas, the traces of the four surface planes on each side the 
centre line on the cutting plane will form a parallelogram, 
the diagonal of which is the trace of the warped surface on 
this cutting plane. Since this cutting plane is any plane par- 
allel to the end areas, and since" the warped surface line bisects 
the figure formed by the two sets of planes formed by the 
diagonals, it follows that the warped surface bisects the volume 
formed by the two sets of planes. The proposition will there- 
fore be established if it be shown that the trace of the warped 
surface is the diagonal of the parallelogram formed by the 
traces of the four planes formed by the two sets of diagonals. 
Fig. 115 shows an extreme case where the centre height is 
higher than the side height at one end and lower at the other. 
Only the left half of the prismoid is shown in the figure. The 



416 



SURVEYING. 



cutting plane cuts the centre and side lines and the two diago- 
nals in efgh on the plane, and in e'f'g'ti on the vertical 
projection. For the diagonal c^ the surface lines cut out are 
e'f and f'h'. For the diagonal c^d x they are e'g' and g'h\ 
For the warped surface the line cut out is e'h' y this being an 




<«W*l) 



Fig. 115. 



cement of that surface. It remains to show that e'f'h'g* is a 
parallelogram. 

Since the cutting plane is parallel to the end planes all the 
lines cut are divided proportionally. That is, if the cutting 
plane is one n th of / from c„ then it cuts off one ;z th of all the 
lines cut, measured from that end plane. But if the lines 
are divided proportionally, the projections of those lines are 
divided proportionally, and hence the points e ',/' ' ,k ' ,g' divide 



THE MEASUREMENT OF VOLUMES. 417 

the sides of the quadrilateral d 9 ', c/,c/,d/ proportionally. But 
it is a proposition in geometry that if the four sides of a quad- 
rilateral, or two opposite sides and the diagonals, be divided 
proportionally and the corresponding points of subdivision 
joined, the resulting figure is a parallelogram. Therefo re e"h' 
g' is a parallelogram, and e'h' is one of its diagonals and hence 
bisects it. Whence the surface generated by this line moving 
along c x c^ and d x d^ parallel to the end areas bisects the volume 
formed by the four planes resulting from the use of both di- 
agonals on one side the centre line. Q. E. D. 

It is probable, therefore, that the warped surface would 
usually fit the ground better than either of the sets of planes 
formed by the diagonals. Furthermore, the errors caused by 
the use of the warped surface (Table X.) are compensating 
errors, thus preventing any marked accumulation of errors in 
a series of prismoids.* There are extreme cases, however, 
such as that given in the example, Fig. 1 14, which are best 
computed by the method by diagonals. 

323. Preliminary Estimate from the Profile. — If the 
cross-sections be assumed level transversely then for given 
width of bed and side slopes, a table of end areas may be pre- 
pared in terms of the centre heights. From such a table the 



* The two methods here discussed are the only ones that have any claims to 
accuracy. The method by " mean end areas," wherein the volume is assumed 
to be the mean of the end areas into the length, always gives too great a volume 
(except when a greater centre height is found in connection with a less total 
width, which seldom occurs), the excess being one sixth of the volume of the 
pyramids involved in the elementary forms of the prismoid. This is a large error 
even in level sections, and very much greater on sloping ground, and yet 
it is the basis of most of the tables used in computing earthwork, and in some 
States it is legalized by statute. Thus in the example computed by Henck's 
method on p. 414 the volume by mean end areas is 1193 cu. yards; by the 
prismoidal formula it is 1168 cu. yards, while by the method by diagonals it was 
only 1001 cu. yards. This was an extreme case, however, and was selected to 
show the adaptation of the method by diagonals to such a form. 
27 



41 8 SURVEYING. 



end areas may be rapidly taken out and plotted as ordinates 
from the grade line. The ends of these ordinates may then 
be joined by a free-hand curve, and the area of this curve 
found by the planimeter. The ordinates may be plotted to 
such a scale that each unit of the area, as one square inch, 
shall represent a convenient number of cubic yards, as iooo. 
The record of the planimeter then in square inches and thou- 
sandths gives at once the cubic yards on the entire length of 
line worked over by simply omitting the decimal point. Evi- 
dently the scale to which the ordinates are to be drawn to give 
such a result is not only a function of the width of bed and 
side slopes, but also of the longitudinal scale to which the pro- 
file line is plotted. The area of a level section is 

A — wc-\-rc\ (i) 

where w, c } and r are the width of base, centre height, and 
slope-ratio respectively. 

Now if h = the horizontal scale of the profile, that is the 
number of feet to the inch, and if one square inch of area is to 
represent iooo cu. yards, the length of the ordinate must be 

hA h {ivc + re 2 ) > , v 

y = = — (2) 

^ iooo X 27 27,000 v J 

If values be given to h, w, and r, which are constants for 
any given case, then the value of y becomes a function of c 
only, and a table can be easily prepared for the case in hand. 
Since y is a function of the second power of c, the second dif- 
ference will be a constant, and the table can be prepared by 
means of first and second differences. Thus if c takes a small 
increment, as 1 foot, then the first difference is 

J 'S = -2i^ W + 2rC + ^ (3) 



THE MEASUREMENT OF VOLUMES. 



419 



But this first difference is also a function of c, and hence when 
c takes an increment this first difference changes by an amount 
equal to 

h 



J 2700 



(4) 



which is constant. An initial first difference being given for a 
certain value of c, a column of first differences can be obtained 
by simply adding the A"y continuously to the preceding sum. 
With this column of first differences the corresponding column 
of values of y may be found by adding the first differences con- 
tinuously to the initial value oi y for that column.* 

TABULAR VALUES OF y IN EQUATION (2) FOR b = 20, s = i£, AND 

h = 400. 



c 


o.'o 


o.'i 


0/2 


o-'3 


0/4 

in. 


o.'5 


0/6 


0/7 


0/8 


o.'g 




in. 


in. 


in. 


in. 


in. 


in. 


in. 


in. 


in. 





0.00 


0.03 


0.06 


0.09 


0.12 


0.15 


o.ip 


0.22 


0.25 


0.28 


I 


•32 


•35 


•39 


.42 


.46 


•49 


•53 


•57 


.61 


.64 


2 


.68 


■72 


.76 


.80 


.84 


.88 


.92 


.96 


1. 00 


1.05 


3 


1. 09 


i-i3 


1. 17 


1.22 


1.26 


i-3* 


i-35 


1.40 


i-45 


1.49 


4 


i-54 


i-59 


1.63 


1.69 


i-73 


1.78 


1.83 


1.88 


i-93 


1.99 


5 


2.04 


2.09 


2.14 


2.19 


2.24 


2.30 


2.36 


2.41 


2.47 


3-52 


6 


2.58 


2.63 


2.69 


2.75 


2.80 


2.87 


2.92 


2.98 


3-°4 


3.10 


7 


3.16 


3.22 


3-28 


3-35 


3-4i 


3-47 


3-54 


3.60 


3.66 


3-73 


8 


3-79 


3-86 


3-92 


3-99 


405 


4-13 


4.19 


4.26 


4-33 


4.40 


9 


4-47 


4-54 


4.60 


4.68 


4-75 


4.82 


4.89 


4-97 


5-°4 


5-" 


10 


5.18 


5-26 


5-33 


5-4o 


5 48 


5-56 


564 


5-7 2 


5-79 


5-8 7 


11 


5-95 


6.03 


6.10 


6.18 


6.26 


6-35 


6-43 


6.51 


6-59 


6.67 


12 


6.76 


6.84 


6.92 


7.00 


7.09 


7.18 


7.26 


7-35 


7-43 


7 -52 


13 


7.61 


7.70 


7.78 


7.86 


7 96 


8.05 


8.14 


8.23 


8.32 


8.41 


14 


8.50 


8.60 


8.68 


8-77 


8.87 


8.97 


9.06 


9.16 


9-25 


9-35 


15 


9.44 


9-54 


9-63 


9-73 


9-83 


9-94 


10.03 


10.13 


10.23 


10.33 


16 


10.43 


10.53 


10.62 


10.73 


10.83 


10.94 


11.04 


11. 15 


11.25 


"•35 


*7 


11.46 


11.56 


11.66 


11.77 


11.88 


12.00 


12.10 


12.21 


12.31 


12.42 


18 


T 2-53 


12.64 


12.75 


12.86 


12.97 


13.09 


13.20 


I3-32 


13-4= 


13-54 


J 9 


13-65 


I 3-77 


1387 


T 3-99 


14.10 


*4 23 


H-34 


14.47 


14.58 


14.70 


20 


14.81 


14-93 


15.04 


15.16 


15.29 


15.42 


J 5-53 


15-66 


15-78 


15.90 



* For a further exposition of this subject, see Appendix C. 



420 SURVEYING. 



The preceding table was constructed in this manner, for 
w = 20 feet, r = i^; and h == 400 feet to the inch. 

324. Borrow-pits are excavations from which earth has 
been " borrowed " to make an embankment. It is generally 
preferable to measure the earth in cut rather than in fill, hence 
when the earth is taken from borrow-pits and its volume is to 
be computed in cut, the pits must be carefully staked out and 
elevations taken both before and after excavating. The meth- 
ods given in art. 311 are well suited to this purpose, or they 
may be computed as prismoids by the aid of Table X„ if pre- 
ferred. To use the table it is only necessary to enter it with 
such heights and widths as give twice the elementary areas 
(triangles or quadrilaterals) into which the end sections are 
divided, and then multiply the final result by the length and 
divide by 100. The table is entered for both end-area dimen- 
sions and also the mid-area dimensions, four times this latter 
result being taken the same as before. 

325. Shrinkage of Earthwork. — Excavated earth first 
increases in volume, when removed from a cut and dumped on 
a fill, but it gradually settles, or shrinks, until it finally comes 
to occupy a less volume than it formerly did in the cut. Both 
the amounts, initial increase, and final shrinkage depend on the 
nature of the soil, its condition when removed, and the man- 
ner of depositing it in place. There can therefore be no gen- 
eral rules given which will always apply. For ordinary clay 
and sandy loam, dumped loosely, the first increase is about one 
twelfth, and then the settlement about one sixth of this increased 
volume, leaving a final volume of about nine tenths of the original 
volume in cut* 

Thus for 100 cubic yards of settled embankment in cubic 
yards in cut would be required. But a contractor should have 

* See paper by P. J. Flynn in Trans. Tech. Soc. of the Pacific Coast, vol. 
ii. p. 179, where all the available experimental data are given. 



THE MEASUREMENT OF VOLUMES. 42 I 

his stakes or poles set one fifth higher than the corresponding 
fill, so that when filled to the tops of these, a settlement of 
one sixth will bring the surface to the required grade. 

These changes of volume are less for sand and more for 
stiff, wet clay. 

For rock the permanent increase in volume is from 60 to 
80 per cent, the greater increase corresponding to a smaller 
average size of fragment. 

326. Excavations under Water. — It is often necessary to 
determine the volume of earth, sand, mud, or rock removed 
from the beds of rivers, harbors, canals, etc. If this be done 
by soundings alone, it is likely to work injustice to the con- 
tractor, as he would receive no pay for depths excavated below 
the required limit ; and besides, foreign material is apt to flow 
in and partially replace what is removed, so that the material 
actually excavated is not adequately shown by soundings 
within the required limits. It is common, therefore, to pay 
for the material actually removed, an inspector being usually 
furnished by the employer to see that no useless work is done 
beyond the proper bounds. The material is then measured in 
the dumping scows or barges. The unit of measure is the 
cubic yard, the same as in earthwork. There are two general 
methods of gauging scows, or boats. One is to actually meas- 
ure the inside dimensions of each load, which is often done in 
the case of rock, and the other is to measure the displacement 
of the boat, which is the more common method with dredged 
material. When the barge is gauged by measuring its dis- 
placement, the water in the hold must always be pumped down 
to a given level, or else it must be gauged both before and after 
loading and the depth of water in the hold observed at each 
gauging. A displacement diagram (or table) is prepared for 
each barge, from its actual external dimensions, in terms of its 
mean draught. There should always be four gaugings taken 
to determine the draught, at four symmetrically located points 



422 SURVEYING. 



on the sides, these being one fourth the length of the barge 
from the ends. Fixed gauge-scales, reading to feet and tenths 
may be painted on the side of the barge, or if it is flat-bot- 
tomed, a gauging-rod, with a hook on its lower end at the zero 
of the scale, may be used and readings taken at these four 
points. Any distortion of the barge under its load, or any 
unsymmetrical loading, will then be allowed for, the mean of 
the four gauge-readings being the true mean draught of the 
boat. 

To prepare a displacement diagram, the areas of the sur- 
faces of displacement must be found for a series of depths uni- 
formly spaced. This series may begin with the depth for no 
load, the hold being dry. They should then be found for each 
five tenths of a foot up to the maximum draught. If the boat 
has plane vertical sides and sloped ends these areas are rec- 
tangles, and are readily computed. If the boat is modelled to 
curved lines, the water-lines can be obtained from the original 
drawings of the boat, or else they must be obtained by actual 
measurement. In either case they can be plotted on paper, 
and their areas determined by a planimeter. These areas are 
analogous to the cross-sections in the case of railroad earth- 
work, and the prismoidal formula may be applied for comput- 
ing the displacement. Thus, 

Let A , A x , A„ A z , etc., be the areas of the displaced water 
surfaces, taken at uniform vertical distances h apart. Then 
for an even number of intervals we have in cubic yards 

V= y^- 2 - 7 ( A o + 4A 1 +2A 2 + 4 A 3 +....A n ). . (i) 

If the total range in draught be divided into six equal por- 
tions, each equal to k, then Weddel's Rule * would give a 



* For the derivation of this rule see Appendix C. 



THE MEASUREMENT OE VOLUMES 423 

nearer approximation. With the same notation as the above 
we would then have, in cubic yards, 

V = 3 ~IA,+A, + A t + A, + $(A,+A, + A t ) + A,l . (2) 



These rules are also applicable to the gauging of reservoirs, 
mill-ponds, or of any irregular volume or cavity. 

After the displaced volume of water is found, the corre- 
sponding volume of earth or rock is found by applying a proper 
constant coefficient. This coefficient is always less than unity, 
and is the reciprocal of the specific gravity of the material. 
This must be found by experiment. In the case of soft mud 
it is nearly unity, while with sand and rock it is much more. 
When rock is purchased by the square yard, solid rock is not 
implied, but the given quality of cut or roughly-quarried rock, 
piled as closely as possible. When rock is excavated, solid 
rock is meant. A measured volume of any material put into a 
gauged scow will give the proper coefficient for that material. 
Thus if the measured volume V give a displacement of V, 

V. 
then -tf = C is the coefficient to apply to the displacement to 

give the volume of that material. 



CHAPTER XIV. 
GEODETIC SURVEYING. 

327. The Objects of a Geodetic Survey are to accurately 
determine the relative positions of widely separated points on 
the earth's surface and the directions and lengths of the lines 
joining them ; or to accurately determine the absolute positions 
(in latitude, in longitude from a fixed meridian, and in eleva- 
tion above the sea-level) of widely separated points on the 
earth's surface and the directions and lengths of the lines join- 
ing them. 

In the first case the work serves simply to supply a skeleton 
of exact distances and directions on which to base a more de- 
tailed survey of the intervening country ; in the second, the re- 
sults furnish the data for computing the shape and size of the 
earth, in addition to their use in more detailed surveys. 

It is usually desirable also to have some knowledge of the 
latitude and longitude of the points determined in the first 
case, but a very accurate knowledge of these would not be es- 
sential to the immediate objects of the work. 

In both cases the points determined form the vertices of a 
series of triangles joining all the points in the system. One or 
more lines in this system of triangles and all of the angles are 
very carefully measured, and the lengths of all other lines in 
the system computed. The azimuths of certain lines are also 
determined, and, if desired, the latitudes and longitudes of some 
of the points. From this data it is then possible to compute 
the latitudes and longitudes of all the points in the system and 



GEODETIC SURVEYING. 425 



the lengths and azimuths of all the connecting lines. The 
work as a whole is denominated triangulation. 

The measured lines are called base-lines, the points deter- 
mined are triangulation-stations, and those points (usually tri- 
angulation-stations) at which latitude, longitude, or azimuth 
is directly determined are called respectively latitude, longi- 
tude, or azimuth stations. The latitude of a station and the 
azimuth of a line are determined at once by stellar observations 
at the point. The longitude is found by observing the differ- 
ence of time elapsing between the transit of a star across the 
meridian of the longitude-station and the meridian of some 
fixed observatory whose longitude is well determined. An ob- 
server at each station notes the time of transit across his merid- 
ian, and each transit is recorded upon a chronograph-sheet at 
each station. This requires a continuous electrical connection 
between the two stations. This difference of time, changed 
into longitude, gives the longitude of the field-station with ref- 
erence to the observatory. 

328. Triangulation Systems are of all degrees of magni- 
tude and accuracy, from the single triangle introduced into a 
course to pass an obstruction, up to the large primary systems 
covering entire continents, the single lines in which are some- 
times over one hundred miles in length. 

The methods herein described will apply especially to what 
might be called secondary and tertiary systems, the lines of 
which are from one to twenty miles in length, and the accu- 
racy of the work anywhere from 1 in 5000 to 1 in 50,000. Al- 
though the methods used are more or less common to all sys- 
tems, yet for the primary systems, where great areas are to be 
covered and the highest attainable accuracy secured, many 
refinements, both in field methods and in the reductions, are 
introduced which would be found useless or needlessly expen- 
sive in smaller systems. 

If it is desired to connect two distant points by a system 



426 



SURVEYING. 



of triangulation at the least expense, then use system I., shown 
in Fig. 1 16. This system is also adapted to the fixing of a 
double row of stations with the least labor. 

If such distant points are to be joined, or such double system 
of stations established, with the greatest attainable accuracy, 
then system III. should be used. This system is also best 
adapted to secondary work, where it is desired to simplify the 
work of reduction. Each quadrilateral is independently re- 
duced. 

If the greatest area is to be covered for a given degree of 
accuracy or cost, then system II. is the one to use. 

System I. consists of a single row of simple triangles, sys- 

i. n. 





Fig. 116. 



tern II. of a double row of simple triangles or of simple tri- 
angles arranged as hexagons, and system III. of a single row 
of quadrilaterals. A quadrilateral in triangulation is an arrange- 
ment of four stations with all the connecting lines observed. 
This gives six lines connecting as many pairs of stations, over 
which pointings have been taken from both ends of the line. 



GEODETIC SURVEYING. 



42; 



For the same maximum length of lines we have the follow- 
ing comparison of the three systems : * 



System. 


Composition. 


Distance 
Covered. 


No. of 
Sta- 
tions. 


Total 
Length 

of 
Sides. 


Area 
Covered. 


No. of 
Conditions. 


I. 

II. 

III. 


Equilateral triangles. 
Hexagons. 
Quadrilaterals (squares). 


5 

5-2 

4-95 


II 

17 
16 


19 

34 
29 


4-5 

9 

3-5 


n — 2 = 9 

7* -14 


5 
in — 4 = 28 



Thus, for the same distance covered, the number of sta- 
tions to be occupied and the total length of lines to be cleared 
out are about one half more for systems II. and III. than for 
system I. The area covered by system II. is twice that by 
system I., but the number of conditions is much greater in 
system III. than in either of the others. Since almost all the 
error in triangulation comes from erroneous angle-measure- 
ments, the results will be more accurate according to our 
ability to reduce the observed values of the angles to their 
true values. The " conditions" mentioned in the above table 
are rigid geometrical conditions, which must be fulfilled (as 
that the sum of the angles of a triangle shall equal 180 ), and 
the more of these geometrical conditions we have, the more 
neatly are we able to determine what the true values of the 
angles are. The work will increase in accuracy, therefore, as 
the number of these conditions increases, and this is why sys- 
tem III. gives more accurate results than systems I. and II. 
This will be made clear when the subject of the adjustment of 
the observations is considered. 

329. The Base-line and its Connections. — The line 
whose length is actually measured is called the base-line. The 



* Taken from the U. S. C. and G. Survey Report for 1876. 



428 



SURVEYING. 



lengths and distance apart of such lines depend on the charac- 
ter of the work and the nature of the ground. Primary base- 
lines are from three to ten miles in length, and from 200 to 



Shot Tower 



CHICAGO 

BASE LINE SYSTEM 

3cale.l: 400,000 




West Base 



Morgan 
-Park 



Fig. 117.* 



600 miles apart. In general, in primary work, the distance 
apart has been about one hundred times the length of the 
base. Secondary bases are from two to three miles in length, 




Lombard ^1___ __i____ ^^ c H IC AGO 
Shot Tower 

\ I ^m 

West Bas^e 

_ ist Bqs 
Willow Springs 



Bald Tom 



Michigan City 
Springvilte 



Otis 



Fig. 118.* 



and from fifty to one hundred and fifty miles apart, the dis- 
tance apart being about fifty times the length of base. Ter- 
tiary bases are from one half to one and a half miles in length 

* Taken from professional papers, Corps of Engineers U. S. Army, No. 24, 
being the final report on the Triangulation of the United States Lake Survey. 



GEODETIC SURVEYING. 429 

and from twenty-five to forty miles apart, the distance apart 
being about twenty-five times the length of base.* 

The location of the base should be such as to enable one 
side of the main system to be computed with the greatest 
accuracy and with the least number of auxiliary stations for a 
given length of base. In flat open country the base may be 
chosen to suit the location of the triangulation-stations in the 
main system ; but in rough country some of the main stations 
must often be chosen to suit the location of the base-line. In 
Fig. 117 the location of the base-line is almost an ideal one, 
being taken directly across one of the main lines of the sys- 
tem. By referring to Fig. 118 it will be seen that the line 
Willow Springs — Shot Tower is one of the fundamental lines 
of the main system, and the base is located directly across it. 
Here the ground is a flat prairie, and the base was chosen to 
suit the stations of the main system. 

The station at the middle of the base is inserted in order 
to furnish a check on the measurements of the two portions 
as well as to increase the strength of the system by increasing 
the number of equations of conditions. Sometimes it is neces- 
sary to use one or more auxiliary stations outside the base 
before the requisite expansion is obtained. Thus suppose the 
stations Morgan Park and Lombard were the extremities of 
the line of the main system whose length was to be computed 
from this base, then the stations Willow Springs and Shot 
Tower might have been occupied as auxiliary stations from 
which the line Morgan Park — Lombard could be computed. 

330. The Reconnaissance. — A system of triangulation 
having been fixed upon, of a given grade and for a given pur- 

* These intervals between bases are in accordance with the practice that 
has hitherto been followed. The new method of measuring base-lines with a 
steel tape, described on p. 450, will probably change this practice by causing 
more bases to be measured, leaving much shorter intervals to be covered by 
angular measurement. 



430 SUR VE YING. 



pose, the first thing to be done is to select the location of the 
base-line and the position of the base-stations. The base should 
be located on nearly level ground, and should be favorably sit- 
uated with reference to the best location of the triangulation- 
stations. These stations are then located, first for expanding 
from the base to the main system, and then with regard to the 
general direction in which the work is to be carried, and to the 
form of the triangles themselves. 

No triangle of the main system should have any angle less 
than 30 nor more than 120 . Although small angles can be 
measured just as accurately as large ones, a given error in a 
small angle, as of one second, has a much greater effect on the 
resulting distances than the same error in an angle near 90 . 
In fact, the errors in distance are as the tabular differences in a 
table of natural sines, for given errors in the angles. These 
tabular differences are very large for angles near o° or 180 , but 
reduce to zero for angles at 90 . The best-proportioned tri- 
angle is evidently the equilateral triangle, and the best-propor- 
tioned quadrilateral is the square. In making the reconnais- 
sance the object should be to fulfil these conditions as nearly 
as possible. 

The most favorable ground for a line of triangles is a valley 
of proper width, with bald knobs or peaks on either side. Sta- 
tions can then be selected giving well-conditioned triangles, 
with little or no clearing out of lines, and with low stations. 
In a wooded country the lines must be cleared out or else very 
tall stations must be used. In general, both expedients are re- 
sorted to. Stations are built so as to avoid the greater portion 
of the obstructions, and then the balance is cleared out. 

So much depends on the proper selection of the stations in 
a system of triangulation, as to time, cost, and final accuracy, 
that the largest experience and the maturest judgment should 
be made available for this part of the work. The form of the 
triangles ; the amount of cutting necessary to clear out the 
lines and the probable resulting damage to private interests ; 



GEODETIC SURVEYING. 43 1 

the height and cost of stations, and the accessibility of the 
same ; the avoidance of all sources of atmospheric disturbance 
on the connecting lines, as of factories, lime- or brick-kilns, and 
the like, which might either obstruct the line by smoke or in- 
troduce unusual refraction from heat ; the freedom from dis- 
turbance of the stations themselves during the progress of the 
work, and the subsequent preservation of the marking-stones — 
these are some of the many subjects to be considered in de- 
termining the location of stations. 

It is the business of the reconnaissance party not only to 
locate the stations, but to determine the heights of the same. 
A station that has been located is temporarily marked by a 
flag fastened upon a pole, and this made to project from the 
top of a tall tree in the neighborhood. In selecting a new 
station it is customary to first select from the map the general 
locality where a station is needed, and then examine the region 
for the highest ground available. When this is found, the 
tallest trees are climbed and the horizon scanned by the aid of 
a pair of field-glasses to see if the other stations are visible. If 
no tree or building is available for this purpose ladders may 
be spliced together and raised by ropes until the desired height 
is obtained. 

331. Instrumental Outfit. — The reconnaissance party re- 
quires a convenient means of measuring angles and of determ- 
ining directions and elevations. For measuring angles a pocket 
sextant would serve very well, provided the stations are distinct 
or provided distinct range-points in line with the stations may 
be selected by the aid of field-glasses. A prismatic pocket- 
compass will often be found very convenient in finding back 
stations which have been located and whose bearings are known. 
An aneroid barometer is desirable for determining approxi- 
mate relative elevations. For methods of using it in such 
work,* see Chapter VI., p. 136. If to the above-named instru- 
ments we add field-glasses, and creepers for climbing trees, the 
instrumental outfit is fairly complete. 



43 2 SURVEYING. 



332. The Direction of Invisible Stations. — It often hap- 
pens that one station cannot be seen from another on account 
of forest growth, which may be cleared out. In such a case the 
station may be located and the line cleared from one station or 
from both, the direction of the line having been determined- 
This direction may always be computed if two other points 
can be found from each of which both stations and the other 
auxiliary point are visible. Thus in Fig. 119 let AB be the 

A _ line to be cleared out, and let C and D 

v ~7\ be two points from which all the stations 

X. / \ may be sighted. Measure the two angles 
yC \ at each station and call the distance CD 

/ X. \ unity. Solve the triangle BCD for the 
/ \j side BC, and the triangle ADC for the 

j /^ -~""^ side AC. We now have in the triangle 
b ABC two sides and the included angle to 

FlG< II9 ' find the other angles. When these are 

found the course may be aligned from either A or B. It will 
often happen that either C or D or both can be taken at regu- 
lar stations. Of course a target must be left at either C or D 
to be used in laying out the line from A or B. The above is a 
modification of the problem given in art. 1 10, p. 107. A use of 
this expedient will often greatly facilitate the work. 

333. The Heights of Stations depend on the relative 
heights of the ground at the stations and of the intervening 
region. If the surface is level, then the heights of stations 
depend only on their distance apart. In any case the dis- 
tance apart is so important a function of the necessary height 
that it is well to know what the heights would have to be for 
level, open country. 

The following table* gives the height of one station when 
the other is at the ground level, for open, level country: 

* Taken from Report of U. S. Coast and Geodetic Survey for 1882. 



GEODETIC SURVEYING. 



433 



DIFFERENCE IN FEET BETWEEN THE APPARENT AND TRUE 
LEVEL AT DISTANCES VARYING FROM i TO 66 MILES. 



Dis- 
tance, 
miles. 

I 


Diffe 


rence in feet for — 


Dis- 
tance, 
miles. 


Difference in feet for — 


Curvature. 


Refraction. 


Curvature 

and 
Refraction. 


Curvature. 


Refraction. 


Curvature 

and 
Refraction. 


0.7 


O.I 


0.6 


34 


771-3 


I08.O 


663.3 


2 


2.7 


0.4 


2-3 


35 


817.4 


II4.4 


703.0 


3 


6.0 


0.8 


5-2 


36 


864.8 


121. I 


743 7 


4 


IO.7 


1 5 


9.2 


37 


913-5 


I27.9 


785.6 


5 


16.7 


2-3 


14.4 


38 


963.5 


134.9 


828.6 


6 


24.O 


3-4 


20.6 


39 


IOI4.9 


142. I 


872.8 


7 


32.7 


4.6 


28.1 


40 


IO67.6 


149-5 


918. 1 


8 


42.7 


6.0 


36.7 


41 


II2I.7 


i57-o 


964.7 


9 


54-o 


7.6 


46.4 


42 


II77.0 


164.8 


IOI2.2 


10 


66.7 


9-3 


57-4 


43 


1233-7 


172.7 


I06I.0 


ii 


80.7 


ir-3 


69.4 


44 


1291.8 


180.8 


Iltl.O 


12 


96.1 


13-4 


82.7 


45 


I35L2 


i8gr.2 


II62.O 


13 


112. 8 


15.8 


97.0 


46 


14H.9 


197.7 


I2I4.2 


14 


130.8 


18.3 


112. 5 


47 


I474.O 


206.3 


I267.7 


15 


150. 1 


21.0 


129. 1 


48 


1537-3 


215.2 


1322. I 


16 


170.8 


23-9 


146.9 


49 


1602.0 


224.3 


1377.7 


17 


192.8 


27.0 


165.8 


50 


1668. 1 


233.5 


1434.6 


18 


216.2 


30.3 


185,9 


5i 


1735.5 


243.0 


1492.5 


19 


240.9 


33-7 


207.2 


52 


1804.2 


252.6 


I55I-6 


20 


266.9 


37-4 


229.5 


53 


I874-3 


262.4 


I6H.9 


21 


94-3 


41.2 


253-1 


54 


1945.7 


272.4 


I673.3 


22 


322.9 


45-2 


277.7 


55 


2018.4 


282.6 


1735.8 


23 


353-0 


49.4 


303.6 


56 


2092 . 5 


292.9 


I799.6 


24 


384-3 


53-8 


330.5 


57 


2167.9 


303.5 


1864.4 


25 


417.0 


58.4 


358.6 


58 


2244.6 


314.2 


I93O.4 


26 


451. 1 


63.1 


388.0 


59 


2322.7 


325.2 


1997-5 


27 


486.4 


68.1 


418.3 


60 


2402 . 1 


336.3 


2065.8 


28 


523-1 


73-2 


449-9 


61 


2482.8 


347-6 


2135.2 


29 


561.2 


78.6 


482.6 


62 


2564.9 


359-1 


2205.8 


30 


600.5 


84.1 


516.4 


63 


2648.3 


370.8 


2277.5 


3i 


641.2 


89.8 


551-4 


64 


2733.0 


382.6 


2350.4 


32 


683.3 


95-7 


587.6 


65 


2819. 1 


394-7 


2424.4 


33 


726.6 


101 .7 


624.9 


66 


2906 . 5 


406.9 


2499.6 



28 



434 SURVEYING. 



square of distance 
Curvature = 



mean diameter of earth ' 
Log curvature = log square of distance in feet — 7.6209807 ; 

Refraction = -77^, where K represents the distance in feet, 

R the mean radius of the earth (log R = 7.3199507), and m the 
coefficient of refraction,* assumed at .070, its mean value, sea- 
coast and interior. 

Curvature and refraction = (1 — 2m) —75. 

v ' 2R 

Or, calling h the height in feet, and K the distance in statute 
miles, at which a line from the height h touches the horizon, 
taking into account terrestrial refraction, assumed to be of the 
same value as in the above table (.070), we have 



757$' ' I 7426* 

The following examples will serve to illustrate the use of 
the preceding table : 

I. Elevation of Instrument required to overcome Curvature 
and Refraction. — Let us suppose that a line, A to B, was 18 
miles in length over a plain, and that the instrument could be 
elevated at either station, by means of a portable tripod, to a 
height of 20 or 30 or 50 feet. If we determine upon 36.7 feet 
at A, the tangent would strike the curve at the distance rep- 
resented by that height in the table, viz., 8 miles, leaving the 
curvature (decreased by the ordinary refraction) of 10 miles to 
be overcome. Opposite to 10 miles we find 57.4 feet, and a 

* See discussion on refraction, under Geodetic Levelling, this chapter. 



GEODETIC SURVEYING. 435 

signal at that height erected at B would, under favorable 
refraction, be just visible from the top of the tripod at A, or 
be on the same apparent level. If we now add 8 feet to tripod 
and 8 feet to signal-pole, the visual ray would certainly pass 6 
feet above the tangent point, and 20 feet of the pole would be 
visible from A. 

II. Elevations required at given Distances. — If it is desired 
to ascertain whether two points in the reconnaissance, esti- 
mated to be 44 miles apart, would be visible one from the 
other, both elevations must be at least 278 feet above mean 
tide, or one 230 feet and the other 331 feet, etc. This sup- 
poses that the intervening country is low, and that the ground 
at the tangent point is not above the mean surface of the 
sphere. If the height of the ground at this point should be 
200 feet above mean tide, then the natural elevations should 
be 478 or 430 and 531 feet, etc., in height, and the line is 
barely possible. To insure success, the theodolite must be 
elevated at both stations to avoid high signals. 

Since the height of station increases as the square of the 
distance, it is evident that the minimum aggregate station 
height is obtained by making them of equal height. Or, if 
the natural ground is higher at one station than the other, 
then the higher station should be put on the lower ground — 
that is, when the intervening country is level. If, however, 
the obstruction is due to an intervening elevation, the higher 
station should be the one nearer the obstruction. 

Sometimes a very high degree of refraction is utilized to 
make a connection on long lines. Thus on the primary trian- 
gulation of the Great Lakes three lines respectively 100, 93, 
and 92 miles in length were observed across Lake Superior, 
which could not have been done except that the refraction was 
found sometimes to exceed twice its average amount. The line 
from station Vulcan, on Keweenaw Point, to station Tip-Top 
in Canada, was 100 miles in length. The ground at station 



43^ SURVEYING. 



Vulcan was 726 feet above the lake, and the observing station 
was elevated 75 feet higher, making 801 feet above the surface 
of the lake. The station at Tip-Top was 1523 feet above the 
lake, the observing tripod being only 3 feet high. From the 
above table we find that the line of sight from Vulcan would 
become tangent to the surface of the lake at a distance of 37.4 
miles, and that from Tip-Top at a distance of 51.5 miles, thus 
leaving a gap of about eleven miles between the points of 
tangency, for ordinary values of the refraction. If this inter- 
val were equally divided between the two stations and these 
raised to the requisite height, we would find from the table 
that Tip-Top would have to be elevated some 340 feet and 
Vulcan some 260 feet. Since this was not done, we must con- 
clude that an occasional excessive value of the refraction was 
sufficient to bend these rays of light by about these amounts 
in addition to the ordinary curvature from this source. In 
other words, the actual refraction when one of these stations 
was visible from the other must have been more than double 
its mean amount. 

The following is a synopsis of the heights of the stations 
built for the observation of horizontal angles in the primary 
triangulation of the Great Lakes : 

Total number of stations * 243 

Combined height of stations 14,100 feet 

Average height of stations 58 " 

Average height of stations from Chicago to Buffalo 81.3 " 

Number of stations less than 10 feet high, . . 22 

from 10 feet to 24 feet in height 18 

" 50 

" 71 

"■ 47 

18 

" • 15 

2 



25 


" 49 


50 


" 74 


75 


" 99 


100 


" 109 


no 


" 119 


120 


" 124 



* Only stations built expressly for the work«are here included. Sometimes 
buildings or towers were utilized in addition to these. 



GEODETIC SURVEYING. 



437 



The heights above given are the heights at which the in- 
strument was located above the ground. The targets were 
usually elevated from 5 to 30 feet higher. 

The excessive heights of the stations from Chicago to 
Buffalo are due to the country being very heavily timbered, 
and the surface only gently rolling. In the vicinity of Lake 
Superior they averaged only about 35 feet high, while from 
Buffalo to the eastern end of Lake Ontario they averaged 51 
feet in height. 

334. Construction of Stations. — If it is found necessary 
to build tall stations, two entirely separate structures must be 



"Rail 



2nd. Splice 




SCAFFOLD 



OBSERVING TRIPOD 



Fig. 



erected, one for carrying the instrument and one for sustain- 
ing the platform on which the observer stands. These should 
have no rigid connection with each other. These structures are 
shown in plan and elevation in Figs. 120 and 121. The inner 
station is a tripod on which the instrument rests ; this is sur- 
rounded by a quadrangular structure, shown separately in ele- 
vation to prevent confusion. Both structures are built entirely 
of wood, the outer one being usually carried up higher than 



438 



SURVEYING. 



the tripod (not shown in the drawing), and the target fixed to 
its apex. This upper framework serves also to support an 
awning to shade the instrument from the sun. For lower sta- 
tions a simpler construction will serve, but the observer's plat- 
form must in all cases be separate from the instrument tripod. 
The wire guys and wooden braces shown in Fig. 120 were not 
used on the U. S. Lake Survey stations. 

For stations less than about 15 feet in height the design 



3~— 
i \ 



--©•- 



\ s / 1 / 

K+f / y 



a ^:—^^__>4 



\/ 



/ ! ^ 



. 1 \ » 



&'- 



/ 



17.68/?. 



17.68 ft. 

ground plan Scale 200 

Fig. 121. 



A 







\ 1 



shown in Figs. 122 and 123 may be used. Here the outer 
platform on which the observer stands is entirely separate from 
the tripod which supports the instrument. For ground stations 
a post firmly planted serves very well, or a tree cut off to the 
proper height. The common instrument tripod will seldom be 
found satisfactory for good work. Sometimes extra heavy and 
stable tripods of the ordinary pattern have given excellent re- 
sults. 

335. Targets. — The requisites of a good target are that it 
shall be clearly visible against all backgrounds, readily bisected, 



GEODETIC SURVEYING. 



439 



rigid, capable of being accurately centred over the station, and 
so constructed that the centre of the visible portion, whetner 
in sun or in shade, shall coincide with its vertical axis. 




Fig. 122. 



It is not easy always to fulfil these conditions satisfactorily. 
To make it visible against light or dark backgrounds, it is well 




Fig. 123. 



to paint it in alternating black and white belts. For ready bi- 
section it should be as narrow as possible for distinctness. This 



440 



SURVEYING. 



is accomplished by making the width subtend an angle of from 
two to four seconds of arc. Since the arc of one second is 
three tenths of an inch for one-mile radius, an angle of four 
seconds would give a target one tenth of a foot in diameter for 
one-mile distances, or one foot in diameter for ten-mile dis- 
tances. Something depends on the magnifying power of 
the telescope used. The design shown in Fig. 124 will satis- 




Fig. 124. 




*^- s CS.-'\^'^J^- ■«•■*•■ r*** "*- atjrS* 



Fig. 12^. 



"VsJ. 



factorily satisfy the conditions as to rigidity and convenience 
of centring. Of course it should stand vertically over the sta- 
tion so that a reading could be taken on any part of its 
height. The last condition is not so easily satisfied. If a 
cylinder or cone be used the illuminated portion only will 
appear when the sun is shining, and a bisection on this portion 
may be several inches to one side of the true axis. 



GEODETIC SURVEYING. 44 1 

The target is then said to present a phase, and corrections 
for this are sometimes introduced. It is much better, however, 
to use a target which has no phase. If the target is to be read 
mostly from one general direction, a surface, as a board, may 
be used ; but if the target is to be viewed from various points 
of the compass, then from those stations which lie nearly in the 
plane of the target it would not be visible, from its width being 
so greatly foreshortened. 

In this case two planes could be set at right angles, one 
above the other. One or both would then be visible from all 
points, and since their axes are coincident, either one could be 
used. The objection to this would be that the upper disk would 
cast its shadow at times on the lower one, leaving one side in 
sun and the other in shade, thus giving rise to the very evil it 
is sought to eliminate. A very satisfactory solution of this 
problem was made on the Mississippi River Survey by means 
of the following device (Fig. 125): Four galvanized-iron wires, 
about three-sixteenths inch in diameter, are bent into a circle of, 
say, four inches in diameter, and soldered. To these four circles 
are attached four vertical wires about one fourth inch in diam- 
eter and four feet long, as shown in the accompanying figure. 
All joints to be securely soldered, the size of the wire increas- 
ing with the size of the target. The target is now divided into 
a number of zones by stretching black and white canvas alter- 
nately and in opposite ways between the opposing uprights, 
making diametral sections. If there are more than two zones, 
those marked by the same color should have the canvas cross- 
ing in different ways, so that if one plane is nearly parallel to 
any line of sight the other plane of this color will be nearly at 
right angles to it. This target has no phase, is visible against 
any background, and readily mounted. A wooden block may 
be inserted at bottom, with a hole in the axis of the target. 
This may then be set over a nail marking the station. The 
target is held at top by wire guys leading off to stakes in the 



442 



SURVEYING. 



ground. Such a target could be mounted on top of the pole 
shown in Fig. 124, if it should be found necessary to elevate it. 
336. Heliotropes. — When the distance between stations 
is such that, owing to the distance, the state of the atmosphere, 
or the small size of the objective used, a target would appear 
indistinct, or perhaps not be visible at all, the reflected rays of 
the sun may be made to serve in place of a target. This limit- 
ing distance is usually about twenty miles. Any device for 
accomplishing this purpose may be called a heliotrope. In 
Figs. 126 and 127 are two forms of such an instrument. That 




Fig. 126. 



Fig. 127. 



shown in Fig. 126 is a telescope mounted with a vertical and 
horizontal motion. This is turned upon the station occupied 
by the observer, and is then left undisturbed. On the tele- 
scope are mounted a mirror and two disks'* with circular open- 
ings. The mirror has two motions so that it can be put into 
any position. Its centre is coincident with the axis of the 
disks, in all positions. The mirror may be turned so as to 

* The disk next to the mirror is unnecessary. 



GEODETIC SURVEYING. 443 

throw a beam of light symmetrically through the forward disk, 
in which position the reflected rays are parallel to the axis of 
the telescope, and hence fall upon the distant point. 

The heliotrope shown in Fig. 127 is to be used in conjunc- 
tion with a single disk, which may be a plain board mounted 
on a plank with the mirror. The silvering is removed from a 
small circle at the centre of the mirror. The disk has a small 
hole through it as high above its base as the clear space on the 
mirror is above the plank. The operator points the apparatus 
by sighting, through the clear spot on the mirror and the open- 
ing in the disk, to the distant station. If the plank be fas- 
tened in this position the attendant now has only to move the 
mirror so as to keep the cone of reflected rays symmetrically 
covering the opening in the disk, and the light will be thrown 
to the distant station. 

Since the cone of incident rays subtends an angle of about 
thirty-two minutes, the cone of reflected rays subtends the 
same angle. The base of this cone has a breadth of about 
fifty feet to the mile distance, or at a distance of twenty miles 
the station sending the reflection is visible over an area in a 
vertical plane 1000 feet in diameter. The alignment of the 
heliotrope need not, therefore, be very accurate. This align- 
ment may vary as much as fifteen minutes of arc on either side 
of the true line. This is nearly 0.01 of a foot in a distance of 
two feet. If the bearing, or direction, of the distant station is 
once determined, it may be marked on the station by some 
means within this limit, and a very rude contrivance used for 
sending the reflected ray, or flash, as it is called. Thus, a mir- 
ror and a disk with the requisite movements may be mounted 
on the ends of a board or pole from five to twenty feet long, 
and when this is properly aligned it serves as well as any other 
more expensive apparatus. The hole in the disk should usually 
subtend an angle at the observer's station of something less 
than one second of arc, which is a width of three-tenths of an 



444 SURVEYING. 



inch to the mile distance. On the best work with large instru- 
ments it should subtend an angle of less than one half a 
second, the minimum effective opening depending almost 
wholly on the condition of the atmosphere.* 

Whatever form of heliotrope is used, an attendant is re- 
quired to operate the apparatus. Evidently it can be used 
only on clear days, whereas cloudy weather is much better 
adapted to this kind of work, since the atmosphere then trans- 
mits so much clearer and steadier an image. 

The heliotrope can be used as a means of communication 
between distant stations by some fixed code of flashing sig- 
nals, and it has been so used very often with great advantage 
to the work. The attendant on the heliotrope, usually called 
a flasher, can thus know when the observer is reading his sig- 
nals, when he is through at that station, and, in general, can re- 
ceive his instructions from his chief direct from the distant 
station. 

337. Station Marks. — If the triangulation is to serve for 
the fixing of points for future reference, then these points must 
be marked in some more or less permanent manner. In this 
case the station has been chosen with this in view, so that if 
possible it has been provided that even the surface for a few 
feet around the station shall remain undisturbed. To insure 
against disturbance from frost or otherwise, the real mark is 
usually set several feet underground. Many different means 
are employed to mark these points. The underground mark 
is to serve only when the superficial marks have been dis- 
turbed, there being always left a mark of some kind projecting 
above ground. On the U. S. Lake Survey, " the geodetic 
point is the centre of a J-inch hole drilled in the top of a stone 

* Reflected sunlight has been seen a distance of sixty miles, through an 
opening one inch in diameter, which then subtended an angle of but one eigh- 
teenth of one second of arc at the instrument. This would require a very clear 
atmosphere. 



GEODETIC SURVEYING. 445 



two feet by six inches by six inches, sunk two and one-half 
feet below the surface of the ground. When the occupation 
of the station is finished, a second stone post, rising eight 
inches above the ground, is placed over the first stone. Three 
stone reference-posts, three feet long, rising about a foot above 
the ground, are set within a few hundred feet of the station, 
where they are the least likely to be disturbed. A sketch of 
the topography within a radius of 400 metres about the sta- 
tion is made, and the distances and azimuths of the reference- 
marks are accurately determined." 

When the station is located in natural rock a copper bolt 
may be set to mark the geodetic point. 

On the Mississippi River survey, stations had to be set on 
ground subject to overflow. These were to serve both for 
geodetic points and for bench-marks, both their geographical 
position and their elevation being accurately determined. 
Both the rank growth and the sedimentary deposits from the 
annual overflows would soon obliterate any mark which was 
but slightly raised above the surface. After much study given 
to the subject, the following method of marking such points 
was adopted : A flat stone eighteen inches square and four 
inches thick, dressed on the upper side, has a hole drilled in 
the centre, into which a copper bolt is leaded, the end project- 
ing a quarter of an inch above the face of the stone. The 

U S 
stone is marked thus, „. *^, and is placed three feet under 

ground. On this stone, and centred over the copper bolt, a 
cast-iron pipe four inches in diameter and five feet long is 
placed, and the dirt tamped in around it. The pipe is large 
enough to admit a levelling-rod. The top is closed with a cap, 
which is fastened to the pipe by means of a bolt. The eleva- 
tions of both the top of the pipe and of the stone are de- 
termined. 




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GEODETIC SURVEYING. 447 

MEASUREMENT OF THE BASE-LINE. 

338. Methods. — The methods heretofore employed in meas- 
uring a base-line have depended on the degree of accuracy 
requisite. If an accuracy of one in one million was desired, then 
the most elaborate primary apparatus has been used, such as 
may be found described in the U. S. Coast and Geodetic Survey 
Reports for 1873 and 1882, or in the Primary Triangulation of 
the U. S. Lake Survey.* For an accuracy of one in fifty 
thousand or one in one hundred thousand, more simple appli- 
ances have been used, such as that shown in Fig. 128. This 
apparatus is fully described and illustrated in the U. S. 
Coast and Geodetic Survey Report for 1880, Appendix 
No. 17. It consists essentially of a four-metre steel bar, 
with zinc tubes on either side of it. One of these zinc tubes 
is attached to the steel bar at one end and the other at the 
other end. Since the expansion of zinc is about two and a 
half times that of steel, it is evident that the corresponding 
ends of the zinc bars will have a relative motion with reference 
to each other as the temperature changes. This relative mo- 
tion is observed by means of the vernier scales attached to the 
ends of the zinc tubes. When the absolute length of the steel 
bar and the coefficients of expansion of both the steel and zinc 
bars are determined, and the readings of the vernier scales for 
a given temperature, then any other temperature will be in- 
dicated by the scale-readings. This combination thus becomes 
a metallic thermometer, from which the temperature of the 
steel rod may be accurately known while in use in the field. 
This assumes that the steel and zinc rods are at the same 
temperature at all times, and that the changes in length due to 
changes in temperature occur simultaneously with the tempera- 

*This is a large quarto volume of 920 pp. and 30 plates, describing the 
methods and results of the geodetic work of the U. S. Lake Survey. It is a 
most valuable contribution to the science of geodesy, and is No. 24 of the 
Professional Papers of the Corps of Engineers of the U. S. Army, 1882. 



448 SUR VE YING. 



ture changes. Unfortunately, this latter condition is not ful- 
filled in the case of zinc. 

From elaborate observations on the relative expansions of 
steel and zinc bars on the United States Lake Survey, it was 
found that zinc is like glass in that its volume-change is not 
wholly coincident with its corresponding temperature-change, 
a residual portion of its change of volume requiring a consid- 
erable time for its completion. In other words, the volume- 
change lags behind the temperature-change, so that its volume 
is not truly indicated by its temperature, it being rather a 
function of the changes in temperature for an indefinite pre- 
vious period. Zinc is, therefore, not a fit metal to use in the 
most accurate measurements, although it is sufficiently reliable 
for a secondary apparatus. 

When two combinations of bars described above are prop- 
erly protected from sun, wind, and from too sudden and varia- 
ble temperature-changes, and when they are mounted in such 
a way as to enable them to be aligned both horizontally and 
vertically, with suitable provision for making exact contacts 
between the ends of the steel bars, they then form a base 
apparatus. 

Sometimes simple wooden or iron rods have been used in 
this way, but then the great source of error is in not knowing 
the mean temperature (and hence length) of the rods at any 
time. If mercurial thermometers are used, these may be many 
degrees warmer or cooler than the bar, since the mercury bulb 
is so much smaller in cross-section than the bar, and therefore 
responds more quickly to changes in temperature. The steel- 
zinc combination is an ideal one, and would be practically per- 
fect if zinc were as reliable a metal as steel. The best metals 
for metallic thermometers are probably steel and brass, the 
coefficient of expansion of the latter being about 1.5 times the 
former.* 

* Mr. E. S. Wheeler, U. S. Asst. Engr., who has had a very large experi- 
perience in the measurement of primary base-lines on the U. S. Lake Survey, 



GEODETIC SURVEYING. 449 

The Steel Tape furnishes the most convenient, rapid, and 
economical means for measuring any distance for any desired 
degree of accuracy up to about one in three hundred thousand, 
and if the most favorable times are chosen, an accuracy of i 
in 1,000,000 may be attained. It is probable, therefore, that 
all engineering measurements, even including primary base- 
lines, will yet be made by the steel tape or by steel and brass 
wires. The conditions of use depend on the accuracy re- 
quired. Let us suppose the absolute length, coefficient of 
expansion, and modulus of elasticity have been accurately 
determined. Any distance can then be measured in absolute 
units within an accuracy of one in one million, by taking due 
precautions as to temperature and mechanical conditions. 
The length of the tape for city work is usually fifty feet, and 
its cross-section about \ inch by -fa inch. That used in New 
York City is -£% inch wide by -fa inch thick. For mining, topo- 
graphical, and railroad surveying a length of one hundred feet, 
with a cross-section of about \ by fa inch, is most convenient. 
For base-line measurement the length should be from three 
hundred to five hundred feet, and its cross-section from two to 
three one-thousandths of a square inch. For an accuracy of 
one in five thousand the tape may be used in all kinds of 
weather, held and stretched by hand, the horizontal position 
and amount of pull estimated by the chainmen. The tempera- 
ture may be estimated, or read from a thermometer carried 
along for the purpose. On uneven ground, the end marks are 
given by plumb-line. 

For an accuracy of one in fifty thousand the mean tem- 
perature of the tape should be known to the nearest degree 
Fahrenheit, the slope should be determined by stretching over 
stakes, or on ground whose slope is determined, and the pull 

recommends the use of a single bar packed in ice, with micrometer microscopes 
mounted on iron stands to mark the end positions of the bar. By this means 
a constant length of standard can be obtained. This has never yet been done, 
however. 

29 



45° SURVEYING. 



should be measured by spring balances. The work could then 
be done in almost any kind of cloudy weather. For an accu- 
racy of one in five hundred thousand, extreme precautions 
must be taken. The mean temperature must be determined 
to about one fifth of a degree F., the slope must be accurately 
determined by passing the tape over points whose elevations 
above a given datum are known, the pull must be known to 
within a few ounces, and all friction must be eliminated. The 
largest source of error is apt to be the temperature. On clear 
days, th*e temperature of the air varies rapidly for varying 
heights above the ground, and, besides, the temperature of the 
tape would neither be that of the air surrounding it, nor of the 
bulb of a mercurial thermometer. In fact, there is no way of 
determining by mercurial thermometer, even within a few 
degrees, the mean temperature of a steel tape lying in the sun, 
either on or at varying heights above the ground. The work 
must then be done in cloudy weather, and ivhen air and ground 
are at about the same temperature. 

There should also be no appreciable wind, both on account 
of its mechanical action on the tape, and from the temperature- 
variations resulting therefrom. 

339. Method, of Mounting and Stretching the Tape. — 
To eliminate all friction, the tape is suspended in hooks about 
two inches long, these being hung from nails in the sides of 
" line-stakes" driven with their front edges on line. These 
stakes may be from twenty to one hundred feet apart. The 
nails may be set on grade or not, as desired ; but if not on 
grade, then each point of support must have its elevation deter- 
mined. A low point should not intervene between two higher 
ones, or the pull on the tape may lift it from this support. 
" Marking-stakes" are set on line with their tops about two feet 
above ground, at distances apart equal to a tape-length, say 300 
feet. Zinc strips about one and one half inches wide are tacked 
to the tops of these stakes, and on these the tape-lengths are 



GEODETIC SURVEYING. 



451 



marked with a steel point. These strips remain undisturbed 
until all the measurements are completed, when they are 
preserved for future reference. In front of the marking-stake 
three " table-stakes" are driven, on which to rest the stretching 
apparatus, and in the rear a " straining-stake" to which to at- 
tach the rear end of the tape. These auxiliary stakes are set 
two or three feet away from the marking-stake, and enough 




Fig. 129. 

lower to bring the tape, when stretched, to rest on the top 
of the marking-stake. 

The stretching apparatus is shown in Fig. 129.* A chain 
is attached to the end of the tape, and this is hooked over the 

_ . — _ - 

* This figure, and the method here described, are taken from the advance- 
sheets of the Report of the Missouri River Commission for 1886. The work 
was in charge of Mr. O. B. Wheeler, U. S. Asst. Engr., who first used this 
method on the Missouri River Survey in 1885. The author had previously 
developed and used the general method, except that he stretched his tape by a 
weight hung by a line passing through a loop which was kept at an angle of 
45° with the vertical, and his end marks were made on copper tacks driven into 
the tops of the stakes. He had also used spring balances for stretching the tape. 



45 2 SURVEYING. 



staple K which is attached to the block KHK ' . This block is 
hinged on a knife-edge at H, and is weighed at K' by the load 
P. The hinge bearing at H is attached to a slide which is 
moved by the screw S working in the nut N. The whole ap- 
paratus is set on the three table-stakes in front of the marking- 
stake, the proper link hooked over the. staple, and the block 
brought to its true position by the screw. This position is 
shown by the bubble L attached to the top of the block. If 
the lever-arms HK and HK' are properly proportioned, the 
pull on the tape is now equal to the weight P. To find this 
length of the arm HK, let HK — k ; HK' = k' ; the horizontal 
distance from the knife-edge H to the centre of gravity of the 
block =£"; and the weight of block = B. 
Then, taking moments about H, we have 

Pk = P& + B#ork = k'+ J ^& . . . . (i) 

When equation (i) is fulfilled then the pull on the tape is just 
equal to the weight P, when the bubble reads horizontally. The 
centre of gravity of the block is found by suspending it from 
two different axes and noting the intersection of plumb-lines 
dropped from these axes. 

At the rear end the tape is held by a slide operated by an 
adjusting screw similar to that shown in Fig. 129. This slide 
rests on the straining-stake, and the rear-end graduation is 
made to coincide exactly with the graduation on the zinc 
strip which marked the forward end of the previous tape- 
length. The rear observer gives the word, and the forward end 
is marked on the next zinc strip. The thermometers are then 
read, and the tape carried forward. 

The measurement is duplicated by measuring again in the 
same direction, the zinc strips being left undisturbed. 

In obtaining a profile of the line the level rod is held on 
the suspension nails and on a block, equal in height to the 
length of the hooks, set on top of the marking-stakes. 



GEODETIC SURVEYING. 453 

For transferring the work to the ground, or to a stone set 
beneath the surface, a transit is mounted at- one side of the 
line and the point transferred by means of the vertical motion 
of the telescope, the line of sight being at right angles to the 
base-line. 

340. M. Jaderin's Method. — Prof. Edward Jaderin, of 
Stockholm, has brought the measurement of distances by 
wires and steel tapes to great perfection. He uses a tape 25 
metres in length, and stretches it over tripods set in line, as 
shown in Fig. 130. On the top of the tripod head is a fixed 
graduation. At the rear end of the tape there is a single grad- 
uation, but at the forward end a scale ten centimetres in length 




Fig. 130. 



is attached to the tape, this being graduated to millimetres on 
a bevelled edge. The middle of this scale is 25 metres from the 
graduation at the other end of the tape. The tripods are set 
as near as may be to an interval of 25 metres, but it is evident 
that the reading may be taken on them if this interval is not 
more than 5 centimetres more or less than 25 metres. The 
reading is taken to tenths of millimetres, the tenths being 
estimated. The tape is stretched by two spring balances, a 
very stiff spring being used at the rear end and a very sensi- 
tive one at the forward end. The rear balance simply tells the 
operator here when the tension is approximately right, the 
measure of this tension being taken on the forward balance, 
which is shown in the figure. v 



454 SURVEYING. 



If a single steel wire or tape be used, Mr. Jaderin also 
finds that the work must be done in cloudy and calm weather, 
or at night, if the best results are to be obtained. But he 
finds that if two wires be used, one of steel and the other of 
brass, he can continue the work during the entire day, even in 
sunshine and wind, and obtain an accuracy of about one in 
one million in his results.* The wires are stretched in succes- 
sion over the same tripods, by the same apparatus, one wire 
resting on the ground while the other is stretched. More ac- 
curate results could doubtless be obtained if both wires are 
kept off the ground constantly, the wire not in use being held 
by two assistants, or if stakes and wire hooks are used, both 
wires might be stretched at once in the same hooks. The two 
wires form a metallic thermometer, the difference between the 
readings of the same distance by the two wires determining 
the temperature of both wires, when their relative lengths at a 
certain temperature and their coefficients of expansion are 
known. This method is similar in principle to that of the 
Coast Survey apparatus, where steel and zinc bars are used, 
shown in Fig. 128. In such cases the true length of line is 
found by equation (5), p. 461. 

At least three thermometers should be used on a 300-foot 
tape, and they should be lashed to the tape or suspended by it at 
such points as to have equal weight on determining its tempera- 
ture. Thus if the tape is 300 feet long the thermometers should 
be fastened at the 50, 150, and 250 foot marks. They should of 
course have their corrections determined by comparison with 
some absolute standard or with other standardized thermom- 
eters. 



* See " Geodatische Langenmessung mit Stahlbanden and metalldrahten," 
von Edv. Jaderin, Stockholm. 1885. 57 pp. Also, " Expose elementaire de 
la nouvelle Methode de M. Edouard Jaderin pour la mesuredesdroitesgeodesi- 
ques au moyen de Bandes d'Acier et de Fils metalliques, "par P. E. Bergstrand, 
Ingenieur au Bureau central d'Arpentage, a Stockholm. 1885. 48 pp. 



GEODETIC SURVEYING. 455 

If the appliances above outlined be used with a single tape 
or wire, and the work be done on calm and densely cloudy 
days, or at night, or with two wires used even in clear weather, 
it is not difficult to make the successive measurements agree to 
an accuracy of one in five hundred thousand. There still re- 
mains, however, the errors in the absolute length, in the coeffi- 
cient of expansion, in the modulus of elasticity, in the measure 
of the pull, and in the alignment, none of which would appear 
in the discrepancies between the successive measurements. 

341. The Absolute Length is the most difficult to deter- 
mine. The best way of finding it would be to compare it with 
another tape of known length. This may be impracticable, 
since all the so-called " standards" are more or less discrepant 
on coming from the makers.* 

If an absolute standard is not available, then the length may 
be found by measuring a known distance, as a previously 
measured base-line, and computing the temperature at which 
the tape is standard. Or the tape may be compared with a 
shorter standard, as a yard or metre bar, by means of a com- 
parator furnished with micrometer microscopes/)- 

* The absolute length of the 300-foot steel tape belonging to the Mississippi 
River Commission, the coefficient of expansion and the modulus of elasticity of 
which the author himself determined in 1880, has now been obtained. This was 
done by measuring a part of the Onley Base Line with this tape, using the 
method herein outlined. This base is situated in Southern Illinois, and forms 
the southern extremity of U. S. Lake Survey primary triangulation-system. The 
probable error in the length of the base, from the original measurements, was 
about one one-millionth. The recent tape-measurements are remarkably accor- 
dant, so the length of this tape is now very accurately known. A similar tape 
belonging to the engineering outfit of Washington University has been com- 
pared with this one at different temperatures, and its absolute length and coeffi- 
cient of expansion found. The 50-foot subdivisions have also been carefully 
determined. 

f Such an apparatus is used in the physical laboratory of Washington Uni- 
versity, which, in conjunction with a standard metre bar which has been com- 
pared with the European standards, enables absolute lengths to be determined 
to the nearest one-thousandth of a millimetre. 



45 6 SURVEYING. 



342. The Coefficient of Expansion may be taken any- 
where from 0.0000055 to 0.0000070 for i° F.* If the tape is 
used at nearly its standard temperature, then the coefficient of 
expansion plays so small a part that its exact value is unim- 
portant. If it is used at a temperature of 70 F. from its 
standard temperature, and if the error in the coefficient used 
be twenty per cent, the resulting error in the work would be 
one in ten thousand. This is probably the extreme error that 
would ever be made from not knowing the coefficient of ex- 
pansion, some tabular value being used. If nothing is known 
of the coefficient of expansion, probably 0.0000065 would be 
the best value to use. It is evident, however, that for the 
most accurate work the coefficient of expansion of the tape 
used must be carefully determined. 

* The author made a series of observations on a steel tape 300 feet long, the 
readings being taken at short intervals for four days and three nights. The 
tape was enclosed in a wooden box, and supported by hooks every sixteen 
feet. The observations were taken on fine graduations made by a diamond 
point, there being a single graduation at one end, but some fifty graduations a 
millimetre apart at the other end. The readings were made by means of 
micrometer microscopes mounted on solid posts at the two ends. The range 
of temperature was about 50 F. , and the resulting coefficient of expansion for 
i° F. was o 00000699 ± 3 in the last place. The coefficient for the Washington 
University tape is 0.00000685. Prof. T. C. Mendenhall found from six or eight 
experiments on steel bands used for tapes, a mean coefficient of 0.0000059. 
Steel standards of length have coefficients ranging from 0.0000048 to 0.0000066. 

Mr. Edward Jaderin, Stockholm, has obtained a mean value of 0.0000055, 
from a number of very careful determinations, both from remeasuring a primary 
base-line, and from readings in a water-bath. Several steel wires were tested, 
and their coefficients all came very near the mean as given above. 

For brass wires he found a mean coefficient of 0.0000096 F. The 15-foot 
standard brass bar of the U. S. Lake Survey has a coefficient of 0.0000100, 
while tabular values are found as high as 0.0000107 F. 

There is some evidence that cold-drawn wires have a less coefficient of expan- 
sion than rolled bars and tapes. 

Coefficients of expansion have seldom been found with great accuracy, the 
coefficients of the " Metre des Archives," the French standard, having had an 
erroneous value assigned to it for ninety years 



GEODETIC SURVEYING. 457 

343. The Modulus of Elasticity is readily found by ap- 
plying to the tape varying weights, or pulls, and observing the 
stretch. The correction for sag will have to be applied for each 
weight used, in case the tape is suspended from hooks, which 
should be done to eliminate all friction. 
Let P x be the maximum load in pounds ; 
P " " minimum load in pounds ; 
a " " increased length of tape in inches due to the 

increased pull ; 
L " " length in inches for pull P , or the graduated 

length of tape ; 
5 " " cross-section in square inches ; 
E " " modulus of elasticity ; 
d " " distance between supports ; 
w " " weight of one inch of tape in pounds ; 
s " " shortening effect of the sag for the length L ; 
sag in inches midway between supports. 



v " 
Then we have 



*=^ 



But for the pull P lf the shortening from sag is much less 
than for the pull P . We must therefore find the effect of the 
sag in terms of the pull. 

344. Effect of the Sag. — Where the sag is small, as it 
always is in this work, the curve, although a catenary, may be 
considered a parabola without an appreciable error. 

If we pass a section through the tape midway between sup- 
ports, and equate the moments of the external forces on one 
side of this section, we obtain, taking centre of moments at 
the support, 

wa d wd* 

~ 2 *4 ~ 8 * 
or 

wd 3 

. v= -*f w 



458 SURVEYING. 



If the length of a parabolic curve be given by an infinite 
series, and if all terms after the second be omitted, which they 

may when , is small, then we may write — 

O 2 

Length of curve = dl i +-^j (2) 

If we now substitute for v its value as given in equation 
(1), we have 

Length of curve = d | 1 + - ~ ^j j- . 

If we call the excess in length of curve over the linear dis- 
tance between supports the effect of the sag, we have 

dfwdV ' 

C = 2 A \PJ (3) 

for one interval between supports. If there are n such inter- 
vals in one tape-length, then nd = L, and the effect of the sag 
in the entire tape-length is 

L iwd\ % 



G--3W (4) 

If S 1 and 5 be the effects of the sag for the pulls P l and P Q 
(S x <S n ior P^P^y then the total movement at the free end 
due to the pull being increased from P to P, would be a-\- 
(S — 5J. If this total movement be called M, then we would 
have 

P _ (P, - P> )L /W>„ ,. 

n ~ S(M-S.+ Sd ~ s [M_{wdf fP:-_P£> 



l 24 v p;p c 



GEODETIC SURVEYING. 459 



Example. 
Let P\ = 60 pounds; 

P = 10 pounds; 

w = 0.00055 pound per inch of tape; 

d = 300 inches = 25 feet; 

S = 0.002 square inch; 

^f = 3.2 inches; 

Z = 3600 inches = 300 feet. 
To find £. 
From equation (5) we have 



50 

= 28,500,000. 



o 002 $ — - - OQ27 ( 35 °° \ 
I 3600 24 \360000y 



From the same data, we find from eq. (4) the effect of the sag to be 0.040 
inch for the ten-pound pull, and 0.001 inch for the sixty-pound pull. 

Evidently, if the tape is stretched by the same weight when its absolute 
length is found, and when used in measuring, the stretch, or elongation from 
pull, would not enter in the computation, and so the modulus of elasticity 
would be no function of the problem. 

Again, the stretch per pound of pull may be observed for the given tape, and 
then neither E nor S, the cross-section, would enter in the computation. 

345. Temperature Correction. — If mercurial thermome- 
ters are used, their field-readings must first be corrected for 
the errors of their scale-reading, each thermometer having, of 
course, a separate set of corrections. Then the mean of the 
corrected readings may be taken for all the whole tape-lengths 
in the line measured, and the correction for the entire line 
obtained at once. Thus, 

let L = length of line ; 

T = temperature at which the length of the tape is given 
for the standard pull P , this usually being the tem- 
perature at which its true length is its graduated 
length for that standard pull ; 
Tm = the mean corrected temperature of the entire line ; 
a — coefficient of expansion for i° ; 
Ct = correction for temperature. 



4^0 SURVEYING. 



Then C, = + *(T m - T a )L (1) 

The temperature correction for a part of a tape-length is com- 
puted separately. 

If the value of a for the tape used is not known, it may be 
taken at 0.0000065. 

If a metallic thermometer is used, as a brass and a steel 
wire, or a brass and a steel bar as in the U. S. C» and G. S. 
apparatus shown on p. 446, then we have the following : 

346. Temperature Correction when a Metallic Ther- 
mometer is used. 

Let / = length of wire or tape used, as 300 feet ; 

4 = absolute length of the steel wire at the standard 

temperature of, say, 32 F. ; 
4 = same for brass wire ; 
L = total length of line for whole tape-lengths ( = nl 

approximately) ; 
n = number of lengths of the standard measured ; 
r s = mean value of all the scale-readings on steel wire 

2r" 



for the entire line I — 

\ n 

r b = same for scale-readings on brass wire ; 

a s = coefficient of expansion for the steel wire ; 

a h — " " " " " brass " 

t == mean temperature for the entire line. 

Then we have 

L = n(l s + r s )(i+(t -32°)a s ) 
= ^(4 + r & )(i+(/ -32> 6 ) 



(2) 



Since the temperature correction is relatively a very small 
quantity, we may put l s + r s = l b + r h = /, the length of the 
tape to which the temperature correction is applied. 



GEODETIC SURVEYING. 



461 



We then have from (2) 



ft - 32°) 



(/. + r.) - (4 + n) 

K a b — O 



(3) 



Substituting this value of the temperature in (2), we obtain 



L = nU. + r.+ 7r ^((l. + r.)-(l t + r b ))]. . (4) 



b — **« 



If we put l 8 -\-r 8 = S s and l h -\- r b = S b , we have 



= „[s b +(S s -S b )-^--_ 



. . . . (5) 



From either of the equations (5) we may compute the length 
of the line as corrected for temperature. If, however, it is 
desired to find the temperature correction separately, in order 
to combine it with the other corrections, we have 



C st = n(S 8 — S b ) 



a* 



(6) 



for the temperature correction to be applied to the measured 
length by the steel wire, or 



C bt = n(S s — S b ) 



a. 



a^ — a. 



(7) 



as the temperature correction to be applied to the measured 
length by the brass wire. 



4^ 2 SUH VE YING. 



These formulae all apply only to the entire tape-lengths. Any 
fractional length would have to be computed separately, or 
else a diminished weight given to their scale-readings in obtain- 
ing the mean values, r s and r h . 

347. Correction for Alignment, both horizontal and ver- 
tical. — The relative elevations of the points of support are 
found by a levelling instrument, and the horizontal alignment 
done by a transit or by eye. An alignment by eye will be 
found sufficiently exact if points be established on line by 
transit every 500 or 1000 feet. The suspending nails and hooks 
afford considerable latitude for lateral adjustment when the 
tape is stretched taut ; hence the horizontal deviation will be 
practically zero unless the stakes are very badly set, and the 
relative elevations of any two successive supports should be 
determined to less than 0.05 foot. If no care is taken to have 
more than two suspension points on grade, then each section 
of the tape will have a separate correction. Usually a single 
grade may as well extend over several sections, in which case the 
portion on a uniform grade may be reduced as a single section. 

Let /,, / 2 , / 3 , etc., be the successive lengths of uniform grades, 
and h iy h„ h 3} etc., the differences of elevation between the 
extremities of these uniform grades ; then for a single grade we 
would have the correction 



C=l- S/P- h* 
or P -2Cl+C=.r-k\ 

m 

But since C is a very small quantity as compared with /, 
we may drop the C*, whence we have C = —, for a single grade. 
The exact value of C y in ascending powers of h, is 

C =-2l+SF + T6T° + etC W 



GEODETIC SURVEYING. 463 

For the entire line, if all but the first term be neglected, 
the correction is 

1 ih; v a; k;\ 

c °=—2{j+j + 7;+---z) (2) 

If the /'s are all equal, as when no two successive suspen- 
sion points fall in a horizontal line, then we have 

1 2A* 

c„=- - t w + v +v+... v> = --&•■ (3) 

Since the relative elevations are determined, and not the 
angles of the grades, these formulae are more readily applied 
than one involving the grade angles. 

The error made in rejecting the second power of C in the 
above equations is given in the table on the following page, 
where / and h are taken in the same unit of length.* 

If the grades are given in vertical angles, as they always 
are with the ordinary base apparatus, then we have for the 
correction to each section whose length is /, and whose grade 
is 6 above or below the horizon, 

e 

C g — — /(i — cos 0) = — 2/ sin 2 -. 

If 6 be expressed in minutes of arc, and if the grade angle 
is less than about six degrees, or if the slope is less than one in 
ten, we may write 

r = - 2/ sin 2 -=_!#» s in a i' = - ^Sll' 07 

2 2 2 

= —0.00000004231 6*1; 

* From Jaderin's Geodatische Langenmessung. 



464 



SUR VE YING. 



or by logarithms, 

log C g = const, log 2.626422 -\- 2 log -j- log /. 



TABLE OF RELATIVE ERRORS IN THE FORMULA C a = 



2/* 



Length of 

Uniform 

Grade. 


t, , • t- Error 
Relative Error = . 


O.OOOO5 


0.00015 

h - R 


0.00025 0.00035 
ise or Fall in Length /. 


0.00045 


I 
2 
3 
4 

5 

6 

7 
8 

9 
10 


O.14 
.24 
■32 
.40 
•47 


O.19 

•31 
.42 

• 53 
.62 




- 






















•54 
.61 

.67 
•73 
•79 


• 71 
.80 

•88 

•97 
1.05 


O.81 

.91 

1.00 

1. 10 

1. 19. 














1. 19 
1.29 


. ... . 


11 
12 
13 
14 
15 


•85 
.91 

•97 
1.02 
1.08 


1. 12 
1.20 
1.27 

i-35 
1.42 


1.28 
1.36 
i-45 
i-53 
1. 61 


i-39 
1.48 

1-57 
1.66 

1-75 






1.67 
1.77 
1.86 


16 

17 

18 

19 
20 


1. 13 
1. 18 
1.24 
1.29 
i-34 


1.49 
1.56 
1.62 
1.69 
1.76 


1.69 

i-77 
1.85 
1.92 
2.00 


1.84 
1.92 
2.01 
2.09 
2.17 


1.96 
2 05 
2.14 
2.23 
2.31 


21 
22 

23 
24 

25 


1-39 
1.44 
1.48 

1-53 

1.58 


1.82 

1.89 

1-95 
2.02 
2.08 


2.07 

2.15 
2.22 
2.29 
2 36 


2.25 

2-33 
2.41 
2.49 
2-57 


2.40 

2.48 

2.57 
2.65 

2.73 


26 

27 

28 

29 

1 30 


1.63 
1,67 
1.72 
1.77 
1. 81 


2.14 
2.20 
2.26 
2.32 
2.38 


2-43 
2.50 

2-57 
2.64 
2.71 


2.65 
2.72 
2.80 
2.87 
2-95 


2.82 
2.90 
2.98 
3.06 
3-14 



GEODETIC SURVEYING. 465 

348. Correction for Sag. — From equation (4), p. 458, we 
have 

C=-T0)'- (4) 

If the standard length be given with the pull P , and the 
distance between supports d , while in the field the pull P and 
distance d between supports be used, then the correction for 
sag is 

^.=f(f-i)=^)'^-^.). • » 

where L, d, and C s are taken in the same unit of length, and w 
is the weight of a unit's length of tape in the same units used 
for P. 

349. Correction for Pull.— From equation (1), p. 457, we 
may write at once 

__ (P-P )L 
Up ~~ T SE * 



Here P is taken in pounds, L and C p in inches, and 5 in 
square inches, since E is usually given in inch-pound units. If 
E has not been determined by experiment, it may be taken at 
28000000. The cross-section S is best found by weighing the 
tape and computing its volume, counting 3.6 cubic inches to 
the pound. Knowing the length, the cross-section can then be 
found. If the stretch has been observed for different weights, 
and the value of E computed, the value of 5 is of no conse- 
quence, provided the same value be used for both observations. 

350. Elimination of Corrections for Sag and Pull. — 
Since the correction for sag is negative and that for pull is 
positive, we may make them numerically equal, and so elimi- 
30 



466 SURVEYING. 



nate them both from the work. If this be done, the normal 
or standard length of the tape should be obtained for no sag 
and no pull, and its normal or standard temperature found such 
that at this temperature, and for no sag and no pull, its gradu- 
ated length is its true length. 

If T is the temperature at which the tape is of standard 
length for the pull P and the distance d between supports, 
and if / is the length of the tape, then we have, 

Shortening from sag = — [—frj , 

PI 

Lengthening from pull == -^~, 

or net lengthening from sag and pull == -~p — —- \-p-j ; 
Lengthening from x degrees F. = xal. 

If, therefore, the effects of sag and pull were eliminated, 
the tape would be of standard length at a temperature x 
degrees above T , where 

-=M-3<f)l <■> 

where all dimensions are in inches and weights in pounds. 

The standard temperature for no sag and no pull would be, 
therefore, 

T n = T + x. . (2) 

t 
We will call this the normal temperature. 



GEODETIC SURVEYING. 467 

In order that the corrections for sag and pull shall balance 
each other, we must have 

EL 1 f^V 
SE~2~4\P1 ' 



SE 

Pn= \/ ~^( wd y> •••••• (3) 

which we will call the normal tension. 

If the stretch in inches is known for one pound of pull for 
the given tape, we may call this e, and we will have 

/ / 

e = ^-~ or SE = -. 
SE e 

Also, Iw = W= weight of entire tape between end graduations, 

W 
or w = -j-. 

I 
And -? == n = number of sags in the tape. 

Substituting these values in (3), we obtain 



where W = weight of entire tape in pounds ; 
/ = length of tape in inches ; 
e = elongation of tape for a one-pound pull ; 

n = number of sags in tape == -3. 

If the tape has no intermediate supports, then n .= I, and 
we have for the normal tension 



3 



P » = \/W (5) 



468 SURVEYING. 



Example. — For the 300-foot steel tape, whose constants the author deter- 
mined, we have W '= 2 lbs., /= 3600 inches, e = 0.066 inch. If the supports 
are 30 feet apart, n = 10, whence, from eq. (4), P n = 4.48 pounds. 

If n = 6, or if the supports were placed 50 feet apart, we would find Pn = 
6.32 pounds. 

If n = 3, or if the supports are 100 feet apart, P n = 10.03 pounds. 

In the last case, the sag would be ten inches midway between supports. 

351. To reduce a Broken Base to a Straight Line. — 
It is sometimes necessary or convenient to introduce one or 
more angles into a base-line. These would never deviate much 
from 180 . Let the difference between the angle and 180 be 
0, and let the two measured sides be a and b, to find the side c. 
If 6 be expressed in minutes of arc and if it is not more than 
about 3 , the following approximate formula will prove suf- 
ficiently exact : 



sin 2 1' abO 
side c = a .+ — 



2 ' a-\-b 
= a-\- b — 0.00000004231 



a-\-b" 



If 6 is greater than from 3 to 5 , the triangle would have to 
be computed by the ordinary sine formula. 

352. To reduce the Length of the Base to Sea-level. 
— In geodetic work, all distances are reduced to what they 
would be if the same lines were projected upon a sea-level 
surface by radii passing through the extremities of the lines. 
It is not necessary, however, to reduce all the lines of a trian- 
gulation system in this manner, since if the length of the base- 
line is so reduced the computed lengths of all the other lines 
of the system will be their lengths at sea-level. The angles 
that are measured are the horizontal angles, and are not affected 
by the differences of elevation of the various stations. It is 



GEODETIC SURVEYING. 469 



necessary, therefore, to know the approximate elevation of the 
base above sea-level. 

Let r = mean radius of earth ; 

a = elevation above sea-level ; 
B == length of measured base ; 
b = length of base at sea-level. 
Then r + a : r : : B : b, 

or b = B 



r -\-d 



The correction to the measured length is always negative, 
and is 

\ r-\-al V -f- &' 

Since a is very small as compared to r f we may write 

C= -B-. 

The mean radius* in feet is 



20026062 + 20855 121 
mean r = — = 20890592 feet, 



log r (in feet) = 7.3199507. 

353. Summary of Corrections. — For the significance of 
the notation used in the following equations, see the preceding 
articles where they are derived, The corrections are all for 



* Rigidly, we should use the length of the normal for the given latitude, but 
the mean radius as above found is sufficient for most cases. 



470 SURVEYING. 

the entire line measured, or rather for that portion of it com- 
posed of entire tape-lengths, and are to be applied with the 
signs given to the measured length. 
I. Correction for Temperature. 
For a single standard with mercurial temperatures, 

i 
C t = + a(T m - T )L (i) 

For metallic thermometer-readings, as found from steel and 
brass standards, for instance, the correction to be applied to 
the length as found by the steel wire, or standard, is 



C st = n(S s -S b )—^— (2) 

a b — a s 

2. Correction for Grade. 

In terms of the difference of elevation of grade, points at a 
common distance, /, apart, 

C o = -tf- • • (3) 



In terms of the grade angles, expressed in minutes of arc, 

C g = — 0.00000)0423 1.2W. (4) 

3. Correction for Sag. 

For the standard length given for a pull P , and a distance 
between supports d , while P and d are used in the field-work, 



GEODETIC SURVEYING. 47 1 

For the standard length given for no pull and no sag, 

c.=-m- < 6 > 

4. Correction for Pull. 

C p =+ P j^L; (7) 

or C p =(P-P„)en (8) 

5. To reduce Standard Temperature to Normal Temperature. 
When the temperature of the tape (T ) is known at which 

the graduated is the true length for the pull P and distance 
between supports d 0J to find the corresponding temperature for 
no pull and no sag, this being called the normal temperature 
(T n ), we have, in degrees, 



T = T 4- - 



\p i (wd \n 

SE 24\/>7j' 



(9) 



6. To eliminate Corrections for Sag and Pull. 



/SE 
Make the pull P n = a / — (wd)* ; (10) 



or P n = A/ (11) 

n y 24en v J 

For no intermediate supports to tape, 



? 3 I vri 

n ~Y 24^' 



(12) 



47 2 SURVEYING. 



P n is called the normal tension. 

7. Correction for Broken Base. 

If a and b are the two measured sides which make an angle 
of 180 — 6, the correction to be added to a -f- b to get the 
distance between their extremities, 6 being less than 5 , and 
expressed in minutes of arc, is 



ab* 
C b = — 0.00000004231 . , . 

8. Correction to Sea-level. 

C--L- 

where L is the length of the measured base at an altitude a 
above sea-level. 

log r (in feet) = 7.3 199507. 

354. To compute any Portion of a Straight Base which 
cannot be directly measured. — It sometimes is convenient 




to take a base-line across a stream or other obstruction to di- 
rect measurement. In such a case a station may be chosen 



GEODETIC SURVEYING. 473 

as O in Fig. 131, and the horizontal angles A OB = P, BOC = 
Q, and COD = R measured. If the parts AB and CD lie in 
the same straight line, and AB = a and CD — b are known, 
then BC — x may be found by measuring only the angles at O. 
Thus in the triangles ABO and ACO we have 

CO x -\-a sin P 



BO~ a sin (P+ Q) J 
also from the triangles BDO and £Z><9 we have 

CO _ £ sin (g + i?) 
j?<? ~~ x -f- b smR ' 

Let K—P-\-Q and £ = Q -f i?, then by equating the 

above values of -^-~ we have 
JdU 

. , .. ab (sin ^T sin Z) 
v ' y v ' J sin -Psin g 



whence 



a 4- b 

* = -—&- ± 



jab (sin iTsin Z) /# — #V 
V sinPsFn"^ ^ \~2~l 



Evidently only the positive result is to be taken. 

The points A, O, and D should be chosen so as to give 
good intersections at A and D. 

355. Accuracy attainable by Steel-tape and Metallic- 
wire Measurements. — The following results have been at- 
tained by using the methods herein described : 

1. In Sweden, Mr. Edw. Jaderin measured a primary 
base-line two kilometres in length three times, by means of 
steel and brass wires 25 metres long, in ordinary summer 



474 SUR VE YING. 



weather, mostly clear, with a probable error of a single deter- 
mination of I in 600,000, and a probable error of the mean re- 
sult of I in 1,000,000, as compared with the true length of the 
line as obtained by a regular primary base apparatus.* 

2. On the trigonometrical survey of the Missouri River, 
in 1885, Mr. O. B. Wheeler, U. S. Asst. Engineer, obtained 
the following results, using one steel-tape 300 feet long: 

Glasgow Base. 

First measurement 7923.237 feet. 

Second " 7923.403 " 

Mean . . 7923.320 ± 0.056 feet. 

In this case the sun was shining more or less on both 
measurements. The probable error of a single result is 1 in 
100,000, and of the mean of two measurements 1 in 140,000. 

Benton Base. 

First measurement *. 9870.443 feet. 

Second " 9870.388 " 

Mean 9870.415 ± 0.018 feet. 

The probable error of a single measurement is 1 in 380,000, 
and of the mean, 1 in 533,000. 

Trovers Point Base. 

First measurement 971 1.915 feet. 

Second " 971 1.892 " 

Mean 971 1.904 ± 0.0078 feet. 

* For title of Mr. Jaderin's pamphlet describing his methods and results, see 
foot-note, p. 454. 



GEODETIC SURVEYING. 475 

The probable error of a single measurement is I in 900,000, 
and of the mean it is 1 in 1,250,000. 

Olney Base. 

First measurement 1082 1.9658 feet. 

Second " 1082 1.9665 " 

Mean 10821.9662 ± 0.0002 feet. 

This base had been measured by the U. S. Lake Survey 
Repsold base apparatus, with a probable error of about I in 
1,000,000. This portion of it, about half the entire base, was 
remeasured with the tape in order to determine the absolute 
length of the tape. The work was done on both the tape- 
measurements in a drizzling rain, so that the temperatures 
were obtained with great accuracy. The mean tempera- 
tures of the two measurements differed, however, by several 
degrees, so that the two sets of graduations on the zinc strips 
were quite divergent, and it was only after the final reduc- 
tion that the two results were known to be so nearly identical.* 

3. The author has measured a number of bases about one 
half mile in length, in connection with students' practice sur- 
veys, by the methods given above, and in each case obtained a 
probable error of the mean of three or four measurements of 
less than one-millionth part of the length of the line. The 
work was always done on densely cloudy days, all the con- 
stants of tape and thermometers being well determined. 



* From advance-sheets of the Report of the Missouri River Commission; 

1886. 



476 



SUR VE YING. 




Fig. 132, 



* GEODETIC SURVEYING. 477 

MEASUREMENT OF THE ANGLES. 

356. The Instruments used in triangulation are designed 
especially for the accurate measurement of horizontal angles. 
This demands very accurate centring and fitting at the axis, and 
strict uniformity of graduation. It was formerly supposed that 
the larger the circle the more accurate the work which could 
be done. It is now known that there is no advantage in having 
the horizontal limb more than ten or twelve inches in diameter. 

There are two general methods of reading fractional parts 
of the angle, smaller than the smallest graduated space on the 
limb. One is by verniers, the other by micrometer micro- 
scopes. Verniers may be successfully used to read angles to 
the nearest ten or twenty seconds of arc, but if a nearer ap- 
proximation is desired microscopes should be employed. 

Fig. 132 shows a high grade of vernier transit, capable also 
of reading vertical angles to 70 . Its horizontal limb is 8 
inches in diameter and reads by verniers to ten seconds. It 
may be used as a repeating * instrument, and used either with 
or without a tripod. To mount such an instrument upon a 
station or post, a trivet, made of brass and shown in Fig. 135, is 
used. The pointed steel legs are driven into the station, the 
centre of the opening being over the station point. The arms 
have angular grooves cut in their upper surface. On this trivet 
may be set any three-legged instrument, so long as the radius 
of its base is not greater than the length <5f the trivet arm's. 

In Fig. 133 is shown a theodolite (not a transit since the 
telescope does not revolve on its horizontal axis) designed for 
the measurement of horizontal angles exclusively. Here mi- 
crometer microscopes are used. The horizontal limb is from 
eight to twelve inches in diameter. There is no vertical circle 
or arc, so that no vertical angles can be read. Since the rela- 
tive heights of triangulation-stations are usually determined 

* See p. -1S4 for explanation of this term. 



478 



SUR VE YING. 




Fig. 133. 



GEODETIC SURVEYING. 



479 




Fig. 134. 



480 SURVEYING. 



from their relative angular elevations or depressions, it is usual- 
ly necessary to have a vertical limb. This instrument could 
not be used as a repeater. 

In Fig. 134 is shown an altazimuth instrument, or an in- 
strument designed for accurately measuring altitudes as well as 
the azimuths of points or lines. Both horizontal and vertical 
limbs are read by means of micrometer microscopes. Such an 
instrument is designed especially for astronomical observations 
for latitude and azimuth, but may also be used as a meridian 

or transit instrument for observ- 
ing time as well as for measuring 
horizontal and vertical angles in 
triangulation. It is in fact the 
universal geodetic instrument, 
just as the complete engineer's 
transit is the universal instrument 
in ordinary surveying. In almost 
all cases where micrometers are 
used in reading the angles the 
limbs are graduated to five or ten minutes and the readings 
made to single seconds. 

357. The Filar Micrometer* is used for the accurate meas- 
urement of small distances or angles, when the required exact- 
ness is greater than can be obtained by means of a vernier 
scale. It is usually combined with a microscope, the microme- 
ter threads and scale lying in the plane of the image produced 
by the objective. This image is always larger than the object 
itself in microscopes, and therefore a given movement of the 
wires in the micrometer corresponds to a very much less dis- 
tance on the object sighted at, according to the magnifying 
power of the objective. 



* From jilum, thread; micros, small, and metros, measure. The thread is in 
this case a spider's web, or scratches on glass. 




GEODETIC SURVEYING. 



481 



The frame holding the movable wires has a screw with a 
very fine thread working in it, called the micrometer screw. 
This screw has a graduated cylindrical head, or disk, attached 
to it, there usually being sixty divisions in the circumference 
when used in angular measurements. The number of whole 
revolutions are recorded by noting how many teeth of a comb- 
scale are passed over, this scale being nearly in the plane of the 
wires and therefore in the focus of the eye-piece. The frac- 
tional parts of a revolution are read on the graduated screw- 
head outside. These micrometer attachments are shown on 
the two microscopes in Fig. 133 and on the five in Fig. 134. 




Fig. 136. 



Fig. 136 is a sectional view of a filar micrometer. The graduat- 
ed head h is attached to the milled head rn, forming a nut into 
which the micrometer-screw a works. This screw is rigidly at- 
tached to the frame b, to which are fastened the movable wires 
f. The comb-scale s and fixed wire f are attached to the 
frame c, which is adjusted to a zero-reading of the graduated 
head by the capstan-screw d. The lost motion on both of 
these frames is taken up by springs. The complete revolutions 
of the screw are counted on the comb-scale, and the fractional 
part of a revolution on the graduated head. The reading is 
made by bringing the double wires symmetrically over a grad- 
uation, the space between the wires being a little more than 
the width of the graduation, when the exact number of revolu- 
tions and sixtieths are read on the comb-scale and on the head. 
31 



4§2 SURVEYING. 



If the limb is graduated to ten minutes and each revolution 
corresponds to one minute, then if the reading is taken on the 
nearest graduation, the number of revolutions need never ex- 
ceed five. If, however, the reading be always taken to the last 
ten-minute mark counted on the limb, then ten revolutions may 
have to be read on the screw. The movement of the threads 
is as they appear to be, there being no inversion of image be- 
tween wires and eye. The movement on the limb is, however, 
opposite from the apparent motion. 

If the limb is graduated to ten minutes, and a single revo- 
lution of the screw corresponds to the space of one minute, 
then just ten revolutions of the screw should move the wires 
from one graduation to the next. If this is not exactly true, 
then the value of a ten-minute space should be measured a 
number of times, by running the wires back and forth, the 
mean result taken, and from this the value of one revolution of 
the screw determined. This value is called the " run of the 
screw," and a correction is applied to the readings, which are 
always made in degrees, minutes, and seconds, counting one 
revolution a minute and one division on the head a second of 
arc. This correction is called "correction for run," and should 
be determined for all parts of the screw used. If the value of 
one revolution is not exactly what it is designed to be, it can 
be adjusted by moving the objective of the microscope in or 
out a little, or the whole microscope up or down with refer- 
ence to the limb, thereby changing the size of the image. 
Even when this adjustment is accurately made, there may be 
still a correction for run on account of the screw-threads not 
being of uniform value. In this case the value of each revolu- 
tion of the screw is determined independently, these values 
tabulated, and the correction for run from this source deter- 
mined for any given reading. Again, as the microscope re- 
volves around the limb with the alidade, the plane of the 
graduations may not remain at a constant distance from the 



GEODETIC SURVEYING. 483 



objective, in which case the size of the image would vary to a 
corresponding degree. To determine this, the values of ten- 
minute spaces are determined on various parts of the limb, 
and if these are not constant, then a table of corrections for run 
may be made out for different parts of the circle. 

For reading on graduated straight lines the double threads 
give better results than either the single thread or the inter- 
secting threads. The space between the threads should be a 
little greater than the width of the image of the graduation- 
line, so that a narrow strip of the limb's illuminated upper 
surface may appear on either side of the graduation and inside 
the wires. The setting is then made so as to make these illu- 
minated lines of equal width. It is conceded that such an ar- 
rangement will give more exact readings than any other that 
has been used. 

The magnifying power of the microscope is from thirty to 
fifty. 

358. Programme of Observations. — There are two gen- 
eral methods of reading angles in triangulation work. One 
method consists in measuring each angle inde- 
pendently, usually by repeating it a number of 
times by successive additions on the limb, and 
then reading this multiplied angle, which is di- 
vided by the number of repetitions to give the 
true value of the angle. In the other method 
the readings are made on the several stations in 
order, as A, B, C, D, and E, in the figure, and 
the angles found by taking the difference between 
the successive readings. Each method has its 
advantages and disadvantages. If the instrument has an ac- 
curate fitting in the axis, clamps which can be set and loosened 
without disturbing the positions of the plates, is provided with 
verniers which have a coarse reading, as twenty or thirty sec- 
onds, and accurate work is desired, and if such an instrument 




484 • SURVEYING. 



is mounted on a low, firm station, then the method by repeti- 
tion would give superior results. If any of these conditions are 
not fulfilled, and especially if the instrument is provided with 
micrometer microscopes, whereby readings may be taken to 
the nearest second of arc, it is much more convenient, cheaper, 
and generally more accurate to read the stations continuously 
around the horizon, back and forth, until a sufficient number 
of readings have been obtained. 

359. The Repeating Method. — This method was for- 
merly used almost exclusively, but the other is the only one 
now used with the most accurate instruments. It was found 
that systematic errors were introduced in the method by 
repetition of a single angle, due largely to the clamping appa- 
ratus. If this method is used the repetitions should be made 
first towards the right and then towards the left ; the number 
of repetitions making a set should be such as to make the mul- 
tiplied angle a multiple of 360 , as nearly as possible, so as to 
eliminate errors of graduation on the limb. Thus, for an angle 
of 6o° repeat it six times and then read. For the second set 
repeat six times in the opposite direction, and with telescope 
inverted. .If triangulation work is to be done with the ordi- 
dary engineer's transit, which reads only to 30 seconds or one 
minute, this method may give very fair results, provided there 
is no movement of circles from the use of the clamping apparatus 
and no lost motion in the axes. The programme would be as 
follows : 

PROGRAMME. 

Telescope Normal, 

1. Set on left station, and read both verniers. 

2. Unclamp above and set on right station. 



3- 


a 


below 


u 


a 


left 


4. 


a 


above 


a 


it 


right 


5- 


<( 


below 


a 


<< 


left 


6. 


<< 


above 
etc., 


<< 


«« 


right 
etc., 



GEODETIC SURVEYING. 485 

until the entire circle has been traversed, then read both ver- 
niers while pointing to right station. The total angle divided 
by the number of repetitions is the measure of the angle 
sought. 

Telescope Reversed. 

1. Set on right station, and read both verniers. 

2. Unclamp above and set on left station. 



3. 


tt 


below " 


(< 


right 


4. 


« 


above " 


<« 


left 


5. 


<t 


below " 


a 


right 


6. 


(( 


above " 
etc., 


it 


left 
etc., 



until the entire circle has been traversed by each vernier, when 
both verniers are read on the left station. 

The repetition in opposite directions is designed to elimi- 
nate errors from clamp and axis movements, and the revers- 
ing of the telescope is designed to eliminate errors arising 
from the line of sight not being perpendicular to the horizon- 
tal axis, and from the horizontal axis not being perpendicular 
to the vertical axis of the instrument.* 

As many such sets of readings may be made as desired, 
but there should always be an even number, or as many of one 
kind as of the other. It will be observed that two pointings 
are taken for each measurement of the angle, but compara- 
tively few readings are made. 

360. Method by Continuous Reading around the Hori- 
zon. — By this method the limb is clamped in any position, and 

* In case the instrument used is a theodolite, and its telescope cannot be 
revolved on its horizontal axis, it should be lifted from the pivot bearings and 
turned over end for end, leaving the pivots in their former bearings. If this 

cannot be done conveniently, then the limb should be shifted by (see next 

n 

page) each time, and this will result in mostly eliminating these same errors of 

collimation and inclination of horizontal axis 



486 SURVEYING. 



left undisturbed except between the different sets of readings. 
The pointings are made to the stations in succession around 
the horizon, and both verniers, or microscopes, read for each 
pointing. Thus, if the instrument were at o, Fig. 137, the 
pointings would be made toA,£, C, D, and E. If the telescope 
is now carried around to the right until the line of sight again 
falls on A, and a reading taken, the observer is said to close 
the horizon ; that is, he has moved the telescope continuously 
around in one direction to the point of beginning. If the two 
readings here do not agree, the error is distributed among the 
angles in proportion to their number, irrespective of their size. 
It is questionable whether such an adjustment adds much to 
the accuracy of the angle values, and therefore it is common 
to read to the several stations back and forth without closing 
the horizon. Sum-angles can afterwards be read if desired. 
Thus, after the regular readings have been taken on the sta- 
tions, the angle AOE, or AOC, and COE, may be read, and so 
one or more equations of condition obtained. 

If the station is tall, there is always a twisting of its top in 
clear weather in the direction of the sun's movement. This 
twisting effect has been observed to be as much as \" in a 
minute of time on a seventy-five-foot station. To eliminate 
this action the readings are taken both to the right and to the 
left. The reading of opposite verniers, or microscopes, elimi- 
nates errors of eccentricity, the inverting of the telescope elimi- 
nates errors of adjustment in the line of collimation and hori- 
zontal axis, and to eliminate periodic errors of graduation each 
angle is read on symmetrically distributed portions of the limb. 
To accomplish this the limb is shifted after each set of read- 

180 
ings an amount equal to ,* where n is the number of sets 

of readings to be taken. The following is the 



* For exception, see foot-note on previous page. 



GEODETIC SURVEYING. 



487 



ist Set. 



PROGRAMME. 

Telescope normal. 

Read to right. 

Read to left. 
Telescope inverted. 

Read to right. 

Read to left. 



Shift the Limb. 



2D Set. * 



Telescope inverted. 

Read to right. 

Read to left. 
Telescope normal. 

Read to right. 

Read to left. 



Shift the Limb. 



Evidently each set is complete in itself, and as many com- 
plete sets may be taken as desired, but no partial sets should 
be used. If the work is interrupted in the midst of one set of 
readings, the partial set of readings should be rejected, and 
when the work is resumed another set begun. In reducing the 
work, if one reading. of any angle is so erroneous as to have to 
be rejected this should vitiate that entire set of readings of 
that angle. 

If preferred, the telescope may be inverted between the 
right and left readings, and then two readings on each mark 
would constitute a complete set, when the limb could be 
shifted again. If this were done, the readings at o, Fig. 137, 
would be : 

c j Telescope Normal — Read ABCDE. 
{ " Inverted " EDCBA. 

Shift the Limb. 

c j Telescope Inverted — Read ABCDE. 
\ " Normal " EDCBA. 

Shift the Limb. 



361. Atmospheric Conditions. — In clear weather not even 
fair results can be obtained during the greater part of the day. 
From sunrise till about four o'clock in the afternoon in sum- 
mer the air is so unsteady from the heated air-currents that 



488 SURVEYING. 



any distant target is either invisible or else its image is so un- 
steady as to make a pointing to it very uncertain. From 
about four o'clock till dark in clear weather, and all day in 
densely cloudy weather with clear air, good work can be done. 
If heliotropes are used, the work is limited to clear weather. 
It has often been proposed to do such work at night, but the 
lack of a simple and efficient light of sufficient strength has 
usually prevented. The higher the line of sight above the 
ground the less it is affected by atmospheric disturbances. 

362. Geodetic Night Signals.— Mr. C. O. Boutelle, of the 
U. S. Coast and Geodetic Survey, made a series of experiments 
in 1879 at Sugar Loaf Mountain, Maryland, for the purpose of 
testing the efficiency of certain night signals and the compara- 
tive values of day and night work. His report is given in Ap- 
pendix No. 8 of the Report of the U. S. C. and G. Survey for 
1880. It seems that either the common Argand or the " Elec- 
tric" coal-oil lamp, assisted by a parabolic reflector or by a 
large lens, gives a light visible for over forty miles. His con- 
clusions are : 

1. That night observations are a little more accurate than 
those by day, but the difference is slight. 

2. That the cost of the apparatus is less than that of good 
heliotropes. 

3. That the apparatus can be manipulated by the same class 
of men as those ordinarily employed as heliotropers. 

4. That the average time of observing in clear weather may 
be more than doubled by observing at night, and thus the time 
of occupation of a station proportionately shortened; "clear- 
cloudy " weather, when heliotropes cannot show, can be utilized 
at night. 

363. Reduction to the Centre. — It sometimes happens 
that the instrument cannot be set directly over the geodetic 
point, as when a tower or steeple is used for such point. In 
tnis case two angles of each of the triangles meeting here may 



GEODETIC SURVEYING. 



489 



be measured and the third taken to be 180 minus their sum, 
or the instrument may be mounted near to the geodetic point 
and all the angles at this station measured from this position. 
These angles will then be very nearly the same as though 
measured from the true position, and may readily be reduced 
to what they would have been if the true station point had 
been occupied. Thus in Fig. 138 let C be the true station to 
which pointings were taken from other stations, and C the posi- 
tion of the instrument for measuring the angles at this station. 
The line AB is a side of the system whose 
length has been found. From the measured 
angles at A and B the approximate value of 
the angle C is found and the lengths of the 
sides a and b computed. At C the angle 
AC B is measured with the same exactness 
as though it were the angle C itself and the 
angle CC ' B = a is measured by a single ob- 
servation. The distance CC = r is also 
found. Since the exterior angle at the inter- 
section Z>, asADB, is equal to the sum of the opposite interior 
angles, we have 




C+y=C'+x, 



or 



C=C + (x-jr). . . (I) 



In the triangle AC'C we have the sides b and r and the 
angle A CC known, whence 



similarly 



sin 


X 


= 


r 


sin 


(C'+ 
b 


> 


sin 


y 


= 


r 


sin 


a 






a 


* 



...... (2) 



Since x and^ are very small angles, their sines are propor- 
tional to their arcs, and we may write sin x = x sin \" where 



490 



SURVEYING. 



x is expressed in seconds ; similarly s'my = y sin i", and equa- 
tions (2) become 

rsm{C-\-a) 



x — 



r sin a 

y = — : Tl . 

Substituting these values in (1) we have 
C 



(3) 



^ + ^P^-^)> • • • (4) 



sin 1 V 



where the correction to C is given in seconds of arc. The 

signs of the trigonometrical 
functions of the angk a must 
be carefully attended to, as it is 
measured continuously around 
to the left to 360 . 

The following is another so- 
lution of the same problem : 
Measure the perpendiculars 
from C upon AC and BC\ Fig. 
139, calling them m and n re- 
"b spectively. Then from equation 
(1) above we have 




Fig. 139 



Fig. 140. 



C=C'+{x-y). 

But since the angles x and y are very small, their sines are 
equal to their arcs, and we have, in seconds of arc, 



m 



x = 



b sin 1 



// 



and 



y — 



n 



# sin i' 



whence 



C = C + -: 



sin I 



m 
K ~b 



(5) 



GEODETIC SURVEYING. 



491 



There are four cases corresponding to the four positions of 
C, as shown in Fig. 140. For these several cases we have 



C 
C 
C 
C 



- 1 ' + sin 1" \b 

c: l 



m n 

a 



= Q + 



m n 

\"\b ~~ a 



m n 
'\J^~ a) '* 



= c; - 



1 fm 

sin \ n \~b ' d 



. . (6) 



ADJUSTMENT OF THE MEASURED ANGLES. 

364. Equations of Conditions. — When any continuous 
quantity, as an angle or a line, is measured, the observed value 
is always affected by certain small errors. Indeed, it would 
not be possible even to express exactly the value of a contin- 
uous quantity in terms of any unit, as degrees or feet and 
fractional parts of the same, even though this value could be 
exactly determined. If, therefore, the measured values of the 
three angles of a triangle be added together, the sum will not 
be exactly 180 . But we know that a rigid condition of all tri- 
angles is that the sum of the three angles is 180 . An equation 
which expresses a relation between any number of observed 
quantities which of geometrical necessity must exist is called 
an equation of condition, or a condition equation. Thus, in 
the above case, if A f , £', and C be the mean observed values 
of the angles, and A, B, and C their true values, we would 
have for our condition equation 



A + B+C= 180 . (1) 



492 



SURVEYING. 



We would also have 

A'+a'=A; B , + b , =B; C' + c'=C, 

where a r , b\ and c' are small corrections to the measured 

values A', B \ and C which are to be 
determined. 

Let us suppose that the length of 
the side b has been exactly meas- 
ured,* then when the true values of 
the angles are found we may com- 
>c pute the other two sides. If the sides 
b and c have both been measured, the 
length of the side c as computed from b must agree with its 
measured length, and so we might write the condition equation 




C = 



b sin (C'+ c 1 ) 
sin (B' + b') ' 



(2) 



Again, if the side a had been measured and its exact length 
found, we would obtain the third condition equation, 



A.= 



b sin (A' + of) 
"sin {B' + V) * 



(3) 



We now have three independent equations involving three 
unknown quantities, and can, therefore, find the quantities a\ 
b\ and c' . But if only one side had been measured, we should 
have had but one equation from which to determine three un- 
known quantities. Evidently there is an infinite number of 

* This assumption is made in regard to the measured base-lines in a trian- 
gulation system, since its exactness is so much greater than can be obtained 
in the angle-measurements. 



GEODETIC SURVEYING. 493 

sets of values of a\ b f , and c', which would satisfy this equation. 
If we now impose the condition that the corrections shall be 
the most probable ones, then there is but one set of values that 
can be taken. 

Equation (i) is called an angle equation, since only angles 
are involved ; while equations (2) and (3) are called side equa- 
tions, since the lengths of the sides are also involved. 

365. Adjustment of a Triangle. — The finding and ap- 
plying of the most probable corrections to the measured values 
of the angles of a system of triangulation is called adjusting 
the system. In the case of a single triangle with one known 
side and three measured angles, we have seen that there is but 
one equation of condition. If the three angles have been 
equally well observed, then it is most probable* that they are 
all equally in error, and hence this condition of highest proba- 
bility gives us the probability equation 

j = V=S (4) 

which enables the corrections to be determined. 

Thus, let A' + B' + C- 180 = a' + V + c' = V, 
then from (4) we have 

e' = *'=,' = |- ( 5 ) 

where V is the error of closure of the triangle. 

* That is, this relation is more probable than are any other single relation 
that can be assigned, but of course it is not more probable than all other cases 
combined. 



494 SURVEYING. 



ADJUSTMENT OF A QUADRILATERAL. 

366. The Geometrical Conditions. — In the quadrilateral 
in Fig. 142 there are eight observed angles, A v B v B z , C 4 , etc. 
The geometrical conditions which must here be fulfilled are : 

(a) The sum of all the angles of any triangle must be 180 
plus the spherical excess* and the opposite angles at the 
intersection of the diagonals must be equal. 

(b) The computed length of any side, as DC, when obtained 
from any other side, as AB, through two independent sets of 
triangles, as ABC, BDC, and ABD, ADC, shall be the same in 
both cases. 

The probability condition is that the set of corrections ap- 
plied to the several angles shall be more probable than any 
other one of the infinite number of sets of corrections which 
would satisfy the other condition. 

The condition given in (a) gives rise to the angle equations, 
and that given in (b) gives one side equations. 

There are evidently eight unknown corrections to be de- 
termined. 

367. The Angle-equation Adjustment. — In the quadri- 
lateral ABCD we have four triangles in which all the angles 
have been observed, two sets of opposite angles where the 
other two angles of the corresponding triangles have been ob- 
served, and the quadrilateral itself in which all the angles have 

*It is not necessary to take account of the spherical excess in computing a 
single triangle or quadrilateral ; but if azimuth is t§ be carried over a series of 
triangles it is necessary that all the angles be spherical angles. In this paper 
spherical excess will be omitted ; but if it is desirable to introduce it, it is in- 
serted in equations (1), (2), and (3), the right members then becoming l v -f- e x% / 2 
-J- e<i, and / 3 -\- e s , where ex is the residual excess of the angle A OB over that of 
the angle DOC (being negative in this case), <? 2 is the excess of angle BOC over 
that of the angle AOD, and e% is the spherical excess for the entire quadri- 
lateral. The spherical excess may be taken as 1" for each 75 square miles of 
area, and this is to be divided equally amongst the angles of the figure. 



GEODETIC SURVEYING. 495 



been observed ; making, in all, seven geometric conditions to 
be fulfilled. Only three of these conditions are independent, 
however, since where any three independent conditions are 
fulfilled the remaining four are fulfilled also. Thus, a great 
variety of conditioned equations could be formed, but we will 




choose the three which give the simplest equations, viz. : that 
the opposite central angles shall be equal, and that the sum of 
all the angles of the quadrilateral shall be 360 . These give 
rise to the following equations : 

I(A lt B„ B a , C v etc., be the observed angles, and /„ / 2 , and / 3 
the residuals in the several combinations, due to erroneous 
determinations, then we have : 

ito°-(A,+BJ - 1 180 - {C t + D,)\ = l„ 

or -A-B, +C. + D, =/,. (1) 

Similarly -£- C t + Z\ + A, = /„ (2) 

and A,+B t +B,+ C t + <T 6 + A + A + A. - 360 = /,. (3) 



496 SURVEYING. 



If the angles have all been equally well observed, — that is, if 
their mean observed values have equal credence, — then they 
are said to have equal weight, and any residual arising from 
any combination of angles should be distributed uniformly 
among the angles forming such combination.* Thus ^ arises 
from the angles A lf B 2 , C b , and£> 6 . This residual should there- 
fore, be divided equally between these four angles. When 
this is done we have 



-A + j-i?, + j + c;-£ + A-^=o, . ( 4 ) 

4 4 4 4 



Similarly 



*+|-ci+J+A-£+^-£=» • (5) 



It is evident that if / s be now divided uniformly among the 
eight observed angles, it will not affect the two adjustments 
already made ; neither have the adjustments already made 
affected the third condition, expressed by eq. (3), since equal 
amounts have been added and subtracted. Hence these ad- 
justments may be made in sequence as well as simultaneously, 
and we shall have for the total corrections for angle-equa- 
tions 

-(§+J)+^-4 + f)-3<5° = °- • • • • (6) 

* The errors in the mean observed values of the angles are supposed to re- 
sult from the small incidental errors and approximations made in pointing, 



GEODETIC SURVEYING. 



497 



Or if v lf v v v„ etc., be the total corrections to the several mean 
observed angles for angle-equations, we have 



v l = v t = 



v z = v k = - 



I, 


-2/, 




8 ' 


/= 


-2/, 



8 



v* = v K = 



v, = v 8 — - 



4 + 2/ 

8 ' 

/ 3 + 2/ 2 

8 ' 



h (7) 



368. The Side-equation Adjustment. — In the quadri- 
lateral shown in the figure, let AB be the known side, and CD 
the required side, which is to be computed through two inde- 
pendent sets of triangles. Let A/, Z? a ', Z? 3 ', etc., be the several 
angles corrected for angle conditions by the corrections found 
in eq. (7). 

As computed through the first set of triangles, we have 



DC = 



BCsm BJ AB sin A/ sin B % ' 



sin Z?.' 



sin C/ sin D/ 



• (») 



c . .. ■ „ r AD sin A/ AB sin £/ sin A/ 

Similarly DC = 



sin CJ 



sin C b f sin D 7 ' 



■ (9) 



Whence 



sin A t 



sin A x r sin B/ sin ,5/ 

sin £7/ sin Z?/ "* sin CJ sin Z>/ ' 



reading, etc. ; in other words, they are supposed to be errors of observation and 
not instrumental errors, these latter having been eliminated by the method of 
making the observations. Since the sources of the errors of observation are 
the same for small as for large angles, it follows that they should be credited 
with equal portions of the aggregate error of any combination of angles, re- 
gardless of the size of the angles themselves. 
32 



49 8 SUR VE YING. 



sin A I sin 2? s ' sin 6"/ sin Z> 7 

sin £/ sin Q' sin Z> 6 ' sin .4/ = If ' ■ ^^ 



r 

'7 

or 



which is called the side-equation. 

It is evident that in any case where the angles have all 
been observed, even after they have been adjusted for the 
angle-conditions, this equation will not hold true, the value of 
the left member being a little more or less than one. When 
put into the logarithmic form for computation, therefore, we 
will have 



log sin A/ -f- log sin B % ' -\- log sin C b r -f- log sin DJ 

—log sin i?/ — log sin C A ' — log sin D/ — log sin AJ = / 4 , (i i) 

where l K is the logarithmic residual due to erroneous observa- 
tions. 

We must now distribute this residual / 4 among the log sines 
according to the most probable manner of the occurrence of 
the errors which caused it. For a given small error, as i" ', in 
any angle, the effect on the log sine is measured by the loga- 
rithmic tabular difference for \" for that angle. This tabular 
difference varies for different angles, being large for angles 
near zero or 180 , and small for angles near 90 . 

Let v/ f v^, v z 'i etc., be the corrections to be made to the 
angles A/, B 2 ', B 3 ', etc, for the side-equation (11), and let d v 
d„ d 3 , etc., be the corresponding logarithmic tabular differences 
for i /r . 

Now, the influences on / 4 of the small angular errors were 
in direct proportion to the tabular differences of the correspond- 
ing log sines ; therefore the corrections should be in proportion 
to the corresponding tabular differences. In other words, the 



GEODETIC SURVEYING. 499 

_ • _ . 

corrections are weighted in proportion to their tabular differ- 
ences.* We therefore have the numerical relation: 

vl : d x :: »,' : d t :: W : d 3J etc., 

or, paying attention to signs, 

But since the log-sine correction is the angular correction 
multiplied by the tabular difference, and since the sum of these 
would equal / 4 , we would have 

v/d.-v/d.+v^d,— ^X+^-^X+z'X— <4= — /*. • (13) 

From equations (12) and (13) we are to find the side-equation 
corrections v/, vj, v 3 ', etc. 

Dividing eq. (13) by eq. (12), term by term, we have 



d l * + d; + d; + d; + d; + d; + d; + d l 



_ '__ hd x _ , l A __ _ l A _ 1 etc 



* To illustrate this principle more fully, let us suppose that for a change 
of 1" in the angles A x and Ai the corresponding changes in the log sines are 1 
for A x and 2 for Ai\ then for a given error of 1 in log sin A x -\- log sin A 2 = I 
there are two chances that it came from A? to one chance that it came from A x , 
when these angles were equally well observed. If the error is to be divided 
between the angles A x and A iy therefore, we should make the correction to A3 

, • . * * «_ V\ v* 

twice as great as the correction to A x , or v x : v% : : d x ' a 2 , whence — — # 

a 1 — di' 

The same reasoning would hold evidently for any number of angles, hence 
equation (12). 



500 SURVEYING. 



Whence we have, for the values of these corrections, 



< 


vi 


< 


v: 


< 


Vl < 


4 ~ 


d, 


~ d,~ 


d t 


~ d t ~ 


d, ""' d, 

v,' l t 
cL ~ 2(d'S 



('4) 



We have now found a set of corrections, v Jf v„ v 3 , etc. (eq. 
7), for the angle-equations, and a set of corrections, #/, z>/, v/ f 
etc. (eq. 14), for the side-equation ; but they were determined 
independently and not simultaneously, and therefore, when 
successively applied, each set of corrections will disturb the 
former adjustment somewhat. Thus, if the corrections in eq. 
(7) be first applied, and then those of eq. (14), using the par- 
tially corrected angles in finding / 4 by eq. (11), we would find 
eq. (10) would be satisfied, but l 19 / 2 , and / 3 , in equations (1), (2), 
and (3), would now not be zero when the newly adjusted angles 
were used. Another set of corrections v", v/ y i/ a ", etc., might 
now be found by eq. (7) for the adjusted angles A", BJ\ B 3 ", 
etc., and so on by successive approximations, using the correc- 
tions of equations (7) and (14) alternately, until both sets of 
conditions were satisfied within the desired limits. It will 
usually be found, however, that the adjustment for side-equa- 
tion does not materially disturb that for smgle-equations. If 
the angles were all the same size, so that the corrections to the 
log sines would have equal weight, the first adjustment would 
remain undisturbed. In this case, the corrections for side- 
equation would all be numerically equal, the odd and even 
subscripts having opposite signs. If the observed angles range 
between 30 and 6o°, as they would in a fairly symmetrical 
quadrilateral, then the errors of this approximation would be 
quite inappreciable. 



GEODETIC SURVEYING. 



$01 



369. Rigorous Adjustment for Angle- and Side-equa- 
tions. — Let the angle-equation adjustments be applied as given 
by eq. (7). Then, using these adjusted angles, let the correc- 
tions to the angles for side- equation be so expressed that they 
shall not be inconsistent with the angle-equation conditions, 
whatever their values. This may be done by letting 






v: = 



**0 1 **"l» 


«V 


■&0 *^1> 


< 


•&0 ~| -^2» 


< 



v: = —x — x. 



2> 



#« 



■^0 "f~ X % » I 
•^0 ** 3 > 

-^o ~r x \ > 



1- • • (15) 



Then, analogous to eq. (13), we may write 
4(*. + *0 — d l x * — *d — d *( x « — *.) + d A (x + * 3 ) 

-f<(^ +^ 3 )-<(^ -^ 3 )-^ 7 (^o-^ 4 )+^o+^4)=-^; • (16) 



or 



{d x — d, — d 3 + d i + d !) — d 6 — d,+ d s )x + <# + dX 

+ (^ + ^K + K + ^K + K + ^sK=-A, 



(17) 



wherein / 4 is given by eq. (11), and the aTs are the tabular 
differences for one second for the several log sines as before. 

If, for simplicity, we write for the coefficients of x , x Jf x at 
x 3 , and;r 4 , respectively, C Q , C„ C 2 , C 6 , and C if then (17) becomes 

£>„ + C x x x + C,x 2 + £> 3 + C<x A = - I,. . . (18) 



It now remains to find the values of x of x lt x„ x 9 , and x if such 
that their combinations which make up the angle-corrections as 
given in eqs. (15) shall be the most probable. 



502 



SURVEYING. 



To make (18) symmetrical with (15), we may put it in the 
following form : 



*. + £>i) + (~ *o + £> 2 ) + (~° * + £> 3 ) 



+ (-^, + £> 4 ) = --/ 4 . , (19) 



By a line of reasoning exactly similar to that by which eq. 
(12) was obtained, we have 



x -f- x x x x x 4x x x 

-~ = -p , whence -~ == y=r. 

—±\r —2 r ° x 

4 +Cl ~4~ 0, 



Similarly with the other equations of (15) and the corre- 
sponding coefficients in (19), from which we may obtain 

C C 3 C 3 C 4 



* As before, the condition of greatest probability makes the corrections pro- 
portional to their coefficients in the fundamental equation (19); hence we have 



Xq : X\ .. — : l-i, 

4 



or, by composition and division, 

'. Xo 

X -f- X\ _ _ Xq — X i 

4 4 

and similarly with x and # 2 . x and x z , etc. 



or 



1 .. ^° 1 r- . Co r . 
x*-\- x x : x — xi :: |- 61 : C\\ 

4 4 



GEODETIC SURVEYING. 503 



Therefore the condition of the highest probability gives 

Co ~ c\ ~ c 7 -~c 3 -c: ( 2 °) 

Dividing (18) by (20), term by term, we have 
C CI CI 

^- + c; + q + c + Q = - — - = - — 

4 4^o *, 

CJ. C£ CJ A 



•*! X '. 



whence 



4-^p x x x^ x a x 4 i 4 



C~C~C~C~C~ C 2 

^0 ^1 ^2 U 3 ^4 ^0 I -SY/-2\ 



(21) 



From equation (21) the side-equation corrections can be 
computed, which will not. disturb the angle-equation adjust- 
ment, and which are the most probable corrections to the 
several angle-values. 

The second or rigid method will be found much more satis- 
factory than the method by approximations. The complete 
adjustment consists in applying to the mean measured 
values, the corrections from angle-equations given by equation 
(7), and then applying to these corrected angles the corrections 
found by equation (21). 

Note. — The results obtained in the above adjustments are 
identical with those found by the method of least squares, and 
the fundamental principle by which they are obtained is also 
the same as that of least squares, viz. : that the arithmetic 



504 SURVEYING, 



mean of properly weighted observations is the most probable 
result, and is identical with that obtained by making the sum 
of the squares of the corrections a minimum. For least-square 
solutions of this problem, see Clarke's " Geodesy," pp. 263-6, 
and Wright's " Adjustment of Observations," pp. 303-8. 

Example. 

The following is the numerical computation of the quadrilateral shown in 
the figure. AB is the known side, and CD is to be found. The mean observed 
values of the angles are given in the second column. The corrections for 
angle-equations are given in the third column, and are the same for all three 
methods of solution given above. The spherical excess is here applied only to 
the quadrilateral as a whole, or to / 3 , thus distributing it equally among the 
several angles. This is a common way of doing it, although if the excess is 
considerable, and the several triangles very unequal in size, as is the case here, 
it should be applied to the several triangles according to their size, as stated in 
the foot-note, p. 495. 

In columns 7 and 8, the corrections for side-equation are worked out by the 
two methods given to show the relative results. Thus, from eq. (14) we find 
the values of v x ' t z/ 3 ', etc., for the first approximation. Applying these to the 
first corrected values in column 4, and again taking out the values of A, / 2 , 
and / 3 , for angle-equation conditions, we find they are not zero, but very small. 
It would probably be sufficient to work out a new set of angle-corrections by 
eq. (7), and then consider the quadrilateral adjusted. In this example, the 
final values thus found would then differ from the final values by the rigid 
adjustment by not more than o".2 for any angle. 

If we compute CD from AB, assuming the latter to be 25,000 feet in length, 
we obtain 88,670.9 ft. in passing through the triangles ABC and BCD, while 
if we pass through the triangles ABD and ADC we obtain 88671. 1 ft., a dis- 
crepancy of 0.2 ft., and giving a mean value of 88671.0 ft. The discrepancy of 
0.2 ft. in the two results by the rigid solution results from not computing the 
corrections beyond tenths of a second. 

If simply a check on the final corrected values is desired, it may be obtained 
by adding them, when their sum should equal 360 -f- spherical excess, or by 
taking out the log sines and seeing if W in eq. (11) is zero. In this case it is 
not zero, but 9, resulting from not carrying out the corrections beyond tenths 
of seconds, as mentioned above. 



GEODETIC SURVEYING. 



505 



H 
Z 
W 
% 
H 

•— > 
Q 
< 

►J 

W 
H 

<i 

Q 
< 

a 



o 
fa 

o 

H 

u 

w 

o 
u 

I 

w 
o 
< 

fa 
o 

o 

H 
< 
H 
D 

o 
u 



eck. 
, Sines 
the 
-ected 
gles. 


tN O 00 VO 





ro 


cn 00 


M 


On 


ON 


on 




a -: 


■* Pi On VO 


->f 


n- VO ON 


T^ 


■* 






2 » 


tN PI VO M 


00 


00 


<*• 


* 


00 


| 


D 




tv w 00 


IN 


tN 


i in 




t^ 










M ON VO CO 
OOOO 


VO 
H 


In CO 00 
VO VO CO 




vo 

H 




a 






N a CO li 





00 


-O f^ 


On 





II 







vcj 


On On O O 


ON 


ON 


On On 


ON 


b> 


in 

c 




CD & 


















*• 




CD 5*o 

*J O Q 


•<f <0 M IT) 

"n vo r->. 




CO 

ON 


^ 

On 10 






8 

8 




CtJ v 









M 








Final Corn 

Angles 

Rigid Solt 

(Third Met 


£rj H eo 

"il Ifl CO Ol 
S >o Tj- Tf 




H 

H 

co 


N 

VO 


W 

VO 
CM 




a 1 




H 
O 


") S + Ul 


OOOO 
-f + + + 


00 ON 00 




in 


ro t-» 


tn 







^ ^? 






CO 00 CO M 




■«*• 


m ro 


CO 


VO 




^ -a 

^-H 3 




!l II II II II 


















& d 




- « CO ■* 






















H H ^ h *t 


















1 


*o „.r 
















"O CN^ 






u 
S 


8 2 -H -j 

DC . w 


vo vo m ■* 




•^-00 co 


VO 







C 

•■a ^ 








c/3 


"b 0" 

+ + + + 





1 




1 1 




1 





Q 


<u 4. 


Q 
O 
I 
H 




Cl 

n 


CO -t-i 

c ctj 


a -h 














O 

X 


•* 


"Si 


-f 


O 3 
















H 


w ^r 




O 1 CO 


33 0* 

cj CD 


on g £ ° 


■* m vo 




cm 


On CI 


Pi 


ro 




8* 5 




.©. ^- 1-^-00 

\j «ni co O 
m ro 


5 


± a-'= ~ 


"b 























I -IIS d 

II II II 


hi 
O 
O 




+ + + 




1 


1 1 


1 


1 


c« 


e ^ 
° 1 

u 1 



u 
















fc 


Vh 


C70 


cl 01 


* u m 


N -t « OO 




co 00 


iri 


M 


or » 



*u 


Tabul 
Diffe 
ence: 


VO O ""> w 

co n vo 

+ + + + 




On 00 tN N 

H 00 PI 

+ + + + 


w 
VO 
CM 































CO t^ >n ■* 


















M 4J 




00 00 vo Pi m 


















^ 




•* On tN 1-1 


















^ CQ 




co t^ pi •<*• 


CO* 

a 

a 


in N vo n 


10 


H 


vo 


m 


O 


10 


NT 






N CO 00 •<!- 
tN pi VO M 


00 
in 


CO 


1- 


* 

* 


On 


10 

H 


+ ^ 




II II II II II 




tN O PI CO 


in 


in 


- r^ 




t^ 








ifi 


m Ov VO CO 


VO 


in 


-0 CO 


r> 


NO 


1 


II rt . 








ON 




vo vo cb 


ON 


M 


1 




c c-t c* & a 


bi 




in on 00 10 





00 


■0 r^ 


O 











O --■ C-* « "«■ 


ON ON ON On 




On 


3n On 


ON 




II 


ivo ^* 




<o O tj (0 Cj 


►J 














10. m - 

IT) tN ** 




^ 
















^r 


1 >-< 

+ c 








First 
Corrected 
Values. 














CO VO H 




in 00 


ON 


0> 




*o 






"w on 10 In 
""> h co 




On On vo 
M pi 


M 









II 

]ct3 
^ US- "° 




p) IT) PI N CO 


Nt m Oi « + 
•«r 10 00 ifl io 


<_> 
















/1 s 




II II II II II 


en 1 


VO ro CO N 




In CO 00 


CM 


On 




1 s< 




Cor- 
rection 

for 
Angle 
equa- 
tions. 


5 














Pi 




r° r rt f CT c™ r* 

nJ vj nJ (j Ij 


1-1 

1 1 1 1 




OOO 
1 1 1 


H 
1 


lO 
1 




M 








**• ro ■*■ co 




■* M 00 


M 


00 










*o 


? 


















w 


4J (/]' 


0) ON VO 00 




O vo 


m 


VO 








& 


S> CJ 


IO H O CO 




Pi co 









On 


'♦ 


>- 3 
(Li- 
en cfl 

" D > 


Pi IT) 00 Ov 




w O vo 


NO 







ON v* 

J IO N 





IN 


M IO •<*• T(- 




CN-i pi 1 


CS 














O 














+ + 4 




00 On 00 




N CO S 


ro 













CO CO C> M 




•* m m 


CO 


vo 








C C3 

ft 

u ~ m 
<u ft 














PO 




II II II 


II 






















j 




CO 

"5c 




: 
















: 


V) 

s 








CJ 

c -r 

V ft 


c 
< 


-1 « rt « 






e 


* 


1* '- 


X 


9 
t/3 




CL> 

Ou 








<; CQ CJ Q 




OQ U 


Q 


< 






in 




19 



506 SURVEYING. 



ADJUSTMENT OF LARGER SYSTEMS. 

370. Used only in Primary Triangulation. — The simul- 
taneous adjustment of all the angles in an extended system of 
triangles with one measured base which is taken as exact, is a 
very complicated problem. The methods of least squares must 
here be applied, so that a discussion of this problem belongs 
rather to a treatise on geodesy than to one on surveying. The 
adjustments of a triangle and of a quadrilateral will be found 
sufficient for all secondary work, or such as is intended to serve 
only for topographical or geographical purposes. Especially 
is this true if the stations be so selected that the observed lines 
will form a series of quadrilaterals. The adjustment of these 
quadrilaterals by the rigid method given above gives nearly 
as good results as could be obtained by reducing the work as a 
single system. For a discussion of the least-square methods 
of adjustment of an extended system of triangles the student 
is referred to " Primary Triangulation of the U. S. Lake Sur- 
vey," being Professional Papers, Corps of Engineers U. S. A., 
No. 24; Report of the U. S. Coast and Geodetic Survey for 
1875; Clarke's Geodesy; and especially to Wright's "Adjust- 
ment of Observations." 

The facility and accuracy with which base-lines may now 
be measured by means of long steel tapes will result in actually 
measuring many more lines than has heretofore been done, and 
so errors from angular measurements will not be allowed to 
accumulate to any great extent. It is not improbable that 
geodetic methods will be materially influenced by this new 
method of accurate measurement. 

371. Computing the Sides of the Triangles.— After the 
angles of the system are adjusted, the sides of the triangles are 
computed by the ordinary sine ratio for plane triangles. If 
the system consist of simple triangles, then one side is known 
and the other two sides computed from it. In this case there 
is no check on the computation except what the computer 



GEODETIC SURVEYING. 507 

carries along with him, or what may be obtained from a dupli- 
cate computation. 

If the system be made up of a series of quadrilaterals, then 
the line which is common to two successive quadrilaterals is 
computed through two sets of triangles from the previous known 
side. Thus if the quadrilateral of Fig. 142 be one of a series, 
the lines in common being AB and CD, then AB is computed 
in duplicate from the previous quadrilateral, and the mean of 
the two results taken. In the triangle ABD compute AD, and 
then in the triangle ADC compute DC] in the triangle ABC 
compute BC, and then in the triangle BCD compute DC again. 
There are thus obtained two independent values of DC, as 
computed from AB. If the adjustment had been exact these 
values would have agreed exactly, but the adjusted angles 
were computed only to the nearest second, or tenth of a second ; 
hence the two values of DC will agree only to a corresponding 
exactness. If the system be composed of quadrilaterals and 
the adjustment be made to the nearest second, then the two 
values of DC would probably differ in the fifth or sixth signifi- 
cant figure. If the adjustment be made to the nearest tenth 
of a second, and a seven-place logarithmic table be used, then 
the two values of DC should begin to differ in the sixth or 
seventh place. Of course the adjusted values are not the true 
values of the angles, but simply the most probable values. If 
the angles were not accurately measured the adjusted values 
may still be considerably in error, but any such errors would 
not prevent the two values of CD from agreeing, since this 
agreement is one of the conditions which the adjustment is 
made to satisfy. The disagreement between the two computed 
values of CD comes only from the inexactness of the computed 
corrections to the angles, an angle, like a length, being an in- 
commensurable quantity, and hence some degree of approxi- 
mation is necessary in its expression. If the true computed 
values of CD differ by more than the amounts above signified, 



508 SURVEYING. 



then it is probable that an error has been made in the com- 
putation. 

LATITUDE AND AZIMUTH. 

372. Conditions. — In the methods here given for obtaining 
latitude, azimuth, and time, the instrument used may either be 
an ordinary field transit mounted on its tripod, or a more elabo- 
rate altazimuth instrument, such as shown in Figs. 132 and 134. 
The accuracy sought is only such as is sufficient for topographi- 
cal or geographical purposes. Both the field methods and the 
office reductions are of the simplest character; but all large 
errors are eliminated, so that the results will be found as accu- 
rate as it is possible to obtain with anything less than the regu- 
lar field astronomical instruments. This higher grade of work 
falls within the sphere of the astronomer rather than of the 
surveyor. 

373. Latitude and Azimuth by Observations on Cir- 
cumpolar Stars at Culmination and Elongation. — When 
latitude and azimuth are to be found to a small fraction of a 
minute, or as accurately as can be read on the instrument used, 
if this be an ordinary field transit, the most convenient method 
is by means of observations on circumpolar stars. The observa- 
tion for latitude is made on such a star when it is at its upper 
or lower culmination, since it is then not changing its altitude, 
and the observation for azimuth is made at elongation, since 
then the star is not changing its azimuth. At these times a 
number of readings may be taken on the star, thus eliminating 
instrumental constants by reversals, since a half hour may be 
utilized for this work without the star sensibly changing its 
position so far as the use it is serving is concerned. Two close 
circumpolar stars have been chosen whose right ascensions 
differ by about five hours and thirty minutes. They therefore 
always give a culmination and an elongation about thirty min- 
utes apart. This is very convenient, since it allows observations 






GEODETIC SURVEYING. 



509 



to be made for latitude and azimuth at one setting with a suf- 
ficient intervening interval to complete one set of observations 
before commencing the next. 

The two stars selected are Polaris (a Ursae Minoris), which is 
of the second magnitude, and 51 Cephei, which is of the fifth 
magnitude. Their relative positions are shown in Fig. 143. 



n\(polaris) 




Fig. 143. 



The position of 5 1 Cephei may be de-scribed with reference 
to the line joining " the pointers," in the constellation of the 
Great Bear, with Polaris. Thus, 51 Cephei is to the right of 
this line, when looking towards the pole-star along the line, at 
a distance of about three times the sun's disk from the line, and 
of about five times the sun's disk from Polaris, in the direction 
of the pointers. 

In case 5 1 Cephei is not visible to the naked eye, as it may 
not be on moonlight nights, or with a slightly hazy atmos- 
phere, it may be found, when near elongation, by the tele- 
scope, as follows : 

Having carefully levelled the instrument, turn upon Polaris. 
When 5 1 Cephei is near its eastern elongation Polaris is near 
its upper culmination, and when near its western elongation 
Polaris is near its lower culmination. To find 51 Cephei at 
eastern elongation, therefore, after taking a pointing on Pola- 



5io 



SUR VE YING. 



ris, lower the telescope (diminish the vertical angle) by about 
one degree (if the time is about twenty minutes before elonga- 
tion), and then turn off towards the east about two and a half 
degrees. This will bring the cross wires approximately upon 
the star. 

To find it at western elongation, simply reverse these angles ; 
that is, increase the vertical angle one degree, and turn off to 
the west two and one half degrees. 

The following table gives the times of the elongations and 
culminations of these two stars for 1885 for latitude 40 , which 
may be used for observing azimuth and latitude. The times 
given are for the nights following the dates named in the first 
column. 

TIMES OF ELONGATION AND CULMINATION, 1885. 
LATITUDE, 40 . 



Date. 




Polaris (a 


Urs. Min.). 




51 Cephei. 


Elon- 




Cul- 






Elon- 






Cul- 








ga- 


Time. 


mina- 


Time. 


ga- 


Time. 


mina- 




Time. 




tion. 




tion. 






tion. 






tion. 






Jan. 1 


W 


I2 h 24 m .6 A.M. 


U 


6 h 29 m 


.9 P.M. 


w 


5 h 48 m 


.3 A.M. 


U 


ii 1 


58 m .6 P.M. 


Feb. 1 


i« 


IO 22 .2 P.M. 


L 


4 25 


.6 A.M. 


41 


3 46 


•4 " 


tt 


9 


56 .7 " 


Mar. 1 


it 


8 31 -8 " 




2 35 


.1 " 


ii 


1 56 


.1 " 


it 


8 


6 .4 " 


April 1 


11 


*6 29 .7 " 




12 33 


.1 " 


(1 


" 54 


.O P.M. 


>t 


*6 


4 -3 " 


May 1 


E 


*4 36 .6 A.M. 




x « 35 


.2 P.M. 


It 


9 55 


•9 ' 


L 


4 


4 .2 A.M. 


June 1 


(1 


2 37 .0 " 




8 33 


•7 " 


tl 


*7 53 


•9 " 


tt 


2 


2 .2 " 


July 1 


i< 


12 39 .0 " 




*6 36 


.2 " 


E 


*6 12 


.6 A.M. 


ti 


12 


4 .2 « 


Aug. 1 


11 


IO 38 .1 P.M. 


U 


4 32 


.8 A.M. 


tt 


4 10 


.8 " 


it 


10 


2 .4 P.M. 


Sept. 1 


11 


8 36 .6 '• 




2 31 


•3 " 


it 


2 9 


.1 " 


tt 


8 


.8 " 


Oct. 1 


11 


*6 38 -9 " 




12 33 


.6 " 


it 


12 11 


•4 " 


tt 


*6 


3 •* " 


Nov. 1 


W 


4 26 .4 A.M. 




10 31 


.7 P.M. 


it 


10 9 


.8 P.M. 


U 


3 


59 .5 A.M. 


Dec. 1 




2 28 .2 " 




8 33 


•5 " 


it 


8 12 


.0 " 


tt 


2 


I .8 " 



* Probably not visible to the naked eye. 



From the above table it is evident that both an elongation 

and a culmination of one of these stars can always be obtained. 

For other days than those given in the table, either inter- 



GEODETIC SURVEYING. 



511 



polate, or find by allowing 3 m -94 for one day, remembering 
that each succeeding day the elongation occurs earlier by this 
amount. 

For other years than 1885, take from the table the time cor- 
responding to the given month and day, and add o m -35 for 
each year after 1885 ; also, 

Add i m if the year is the second after leap-year. 

Add 2 m if the year is the third after leap-year. 

Add 3 m if the year is leap-year before March 1. 

Subtract i m if the year is leap-year after March 1. 

For the first year after leap-year there is no correction ex- 
cept the periodic one of o m .35 per annum. 

For other latitudes than 40 , add o m .i4 for each degree 
south of 40 north latitude, or subtract O m . 18 for each degree 
north of 40 north latitude for Polaris, and o m .29 and o m .39 for 
the corresponding correction for 51 Cephei. 

The following table gives the pole distances of Polaris and 
51 Cephei for Jan. 1 of each third year from 1885 to 1930: 





POLE DISTANCE (90 - 


Decli-na 


tion). 






Star. 


18S5. 


1888. 


1891. 


1894. 


1897. 


1900. 


1903. 


1906. 


Polasis 

51 Cephei.. 


i°i8'i6" 
2 46 35 


i i 7 'i9" 
2 4647 


I°l6'2 3 " 

2 47 00 


I°I 5 '26" 

2 47 13 


i°i4 / 3o" 
2 47 26 


i°i3 / 33" 
2 4740 


i°i2 / 37 // 
2 47 54 


i ii' 4 i" 
2 48 09 




Star. 


1909. 


1912. 


1915. 


1918. 


1921. 


1924. 


1927. 


1930. 


Polaris 

51 Cephei.. 


i°io'4 5 " 
2 48 24 


i° g'49" 
2 48 39 


i° 8'5 3 " 
2 48SS 


i° 7 '58" 
2 49 12 


i° 7' 2" 
2 49 28 


i° 6' 7" 
2 49 45 


1° 5 'l2" 

2 50 01 


i* 4 'i6" 
2 50 18 



To find the pole distance for any intermediate time, make 
a linear interpolation between the two adjacent tabular values. 



512 



SUR VE YING. 



To observe for latitude no knowledge of the geographical 
position is needed. 

374. The Observation for Latitude consists simply in 
observing the altitude of a circumpolar star at upper or lower 
culmination and correcting this altitude for the pole distance 
of the star and for refraction. 



Let 



== latitude ; 
d = polar distance ; 
r = refraction ; 
h == altitude ; 



then 



<p = k^f d — r; 



(1) 



the minus sign being used for upper, and the plus sign for 
lower, culmination observations. The value of r is taken from 
the following table of mean refractions computed for barometer 
30 inches, and temperature $o° F. 



TABLE OF MEAN REFRACTIONS. 



Altitude. 


Refraction. 


Altitude. 


Refraction. 


IO° 


5 19 


20° 


2' 39" 


II 


4 51 


25 


2 04 


12 


4 28 


30 


I 41 


13 


4 07 


35 


I 23 


14 


3 50 


40 


I 09 


15 


3 34 


45 


O 58 


16 


3 20 


50 


4y 


17 


3 08 


60 


34 


18 


2 58 


70 


21 


*9 


2 48 


80 


10 



GEODETIC SURVEYING. 513 

The index error of the vertical circle is eliminated by read- 
ing with the telescope direct and reversed, providing the verti- 
cal circle is complete. If the vertical limb is but an arc of 180 
or less, the index error cannot be eliminated in this way. In 
this case the second method is recommended. 

375. First Method. — Mount the instrument firmly, pre- 
ferably on a post, and adjust carefully the plate-bubbles, 
especially the one parallel to the plane of the vertical circle. 
About five or ten minutes before the star comes to its culmi- 
nation read the altitude of the star with telescope direct. 
Revolve the telescope on its horizontal axis and also on its 
vertical axis, relevel the instrument if the bubbles are not in the 
middle, but do not readjust the bubbles, and bring the tele- 
scope upon the star. Make two readings in this position. 
Revolve the telescope and instrument again about their axes, 
relevel, and read again in first position. This gives two direct 
and two reversed readings taken in such a way as to eliminate 
the error from collimation, the index error of vertical circle, 
and also the error of adjustment of the plate-bubbles. The 
result, when corrected for refraction and the pole distance of 
the star, should be the latitude of the place within the limits 
of accuracy and exactness of the vertical circle-readings. 

376. Second Method. — An " artificial horizon," formed by 
the free surface of mercury in an open vessel, may be used in 
conjunction either with the transit or a sextant. If the former 
is used two pointings are made — one to the star and the other 
to its image in the mercury surface. The angle measured is 
then twice the apparent altitude of the star. The position of 
the vessel of mercury will be on a line as much below the 
horizontal as the star is above it. The instrument is first set 
up and then the artificial horizon put in place. The surface 
of the mercury must be free from dust. If the mercury is not 
clean it may be strained through a chamois-skin or skimmed 
by a piece of cardboard. Any open vessel three or more 

33 



5H SURVEYING. 



inches in diameter may be used for holding the mercury. It 
should be placed on a solid support and protected from the 
wind. 

The observations with a transit would then consist in taking 
a reading on the star just before culmination, two readings on 
the image, and then one on the star. The index error of the 
vernier on the vertical circle will then be eliminated, since both 
plus and minus angles have been read, and their sum taken for 
twice the altitude of the star. This method is adapted to 
transits with incomplete vertical limbs. 

The Sextant may also be used with the artificial horizon 
and will give more accurate results than can be obtained with 
the ordinary field transits. The double altitude angle is then 
measured at once by bringing the direct and reflected images 
of the star into coincidence. In both cases the observed angle 
is 2k, and the latitude is found from equation (i), as before. 
If there is much wind the mercury basin may be partially 
covered, leaving only a narrow slit in the vertical plane through 
instrument and star, or the regular covered mercurial horizon 
may be used. This is covered by two pieces of plate-glass set 
at right angles to each other like the roof of a house. If the 
opposite faces of these glasses are not parallel planes, an error 
is introduced. This is eliminated by reversing the horizon 
apparatus on half the observations. It is best, however, to 
avoid the use of glass covers, if possible. 

If tin-foil be added to the mercury an amalgam is formed, 
whose surface remains a perfect mirror, which is not readily 
disturbed by wind. As much tin-foil should be used as the 
mercury will unite with. Observations may then be made in 
windy weather without the aid of a glass cover. 

377. Correction for Observations not on the Meridian. 

If the star is more than five or ten minutes of time from the 

meridian, it is necessary to apply a correction to the observed 
altitude to give the altitude at culmination. The following 



GEODETIC SURVEYING. 5 1 5 

approximate rule gives these corrections for the two circum- 
polar stars here used, with an error of less than \" of arc when 
the observation is taken not more than 18 minutes of time 
from the star's meridian passage, and the error is less than io" 
of arc when the observation is made 32 minutes of time from 
the meridian. 

Rule for reducing cir cummer idian altitudes to the altitude at 
culmination. 

For Polaris : Multiply the square of the time from meridian 
passage, in minutes, by 0*0444, and the product is the correc- 
tion in seconds of arc. 

For 51 Cephei : Multiply the square of time from meridian 
passage, in minutes, by 0.1143, and the product is the correc- 
tion in seconds of arc. 

The correction is to be added to the observed altitude for 
upper culmination, and subtracted for lower culmination. 

By using these corrections an observation for latitude may 
be made at any time for a period of about one hour, near the 
time of culmination. 

378. The Observation for Azimuth is made on one of 
the two stars here chosen when it is at or near its eastern or 
western elongation, for the same reason that latitude observa- 
tions are taken at culmination. The azimuth of a star at 
elongation is found from the formula, 



. . , sine of polar distance 

sine of azimuth = : 7-.- . , — . . . (1) 

cosine of latitude v ' 



This formula is so simple that it is hardly necessary to give a 
table of values of azimuths for various latitudes. Such a table 
is given for Polaris, however, on p. 33. The pole distances 
are given on p. 511, and the latitude is found by observation. 
.It is not necessary to know the azimuth of the star at elonga- 



5 i6 



SURVEYING. 



tion before making the observations. This can be computed 
afterwards from the observed latitude. 

The observation for azimuth consists simply in measuring 
the horizontal angle between the star and some conveniently 
located station, marked by an artificial light. The operation 
is in no sense different from the measurement of the horizontal 
angle between two stations at different elevations. The great 
source of error is in the horizontal axis of the telescope. If 
this is not truly horizontal then the line of sight does not de- 
scribe a vertical plane, and since the two objects observed have 
very different elevations, the angle measured will not be that 
subtended by vertical planes passing through the objects and 
the axis of the instrument. To eliminate this error the tele- 
scope is reversed, and readings taken in both positions. The 
following programme is recommended : 



PROGRAMME FOR OBSERVING FOR AZIMUTH ON A CIRCUM- 
POLAR-STAR AT ELONGATION. 



Instrument. 



Direct. . . 

Reversed 
i « 

«< 

Direct . . . 
<< 

(< 

Reversed 



Time of Observation. 



io min. before elongation, 
7 
4 

2 

2 min. after 
4 
7 
io 



Reading on 



Mark. 
<« 

Star. 



Mark. 



The instrument should always be relevelled after reversing, 
but the bubbles should not be readjusted after the observa- 
tions have begun. If this be done and the above programme 
followed, all instrumental errors will be eliminated except. 



GEODETIC SURVEYING. 517 

those of graduation. Of course both verniers are to be read 
each time. 

Having found the latitude, the azimuth of the star at elon- 
gation is found from equation (1) above. This is then added 
to or subtracted from the horizontal angle between mark and 
star, as the case may be, to give the azimuth of the mark from 
the north point. If the azimuth is to be referred to the south/ 
point, which it generally is, we must add or subtract 180 . 

379. Corrections for Observations near Elongation. — 
As in the case of observations for latitude, we may have an, 
approximate rule for reducing an observed azimuth when near 
elongation to what it would have been if taken at elongation. 
The limits of accuracy are also about the same, but the factors 
are slightly different. 

Rule for reducing azimuth observations on Polaris and 5 1 
Cephei near elongation to their true values at elongation, for 
latitude 40 . 

For Polaris, multiply the square of the time from elonga- 
tion in minutes by 0.058, and the product will be the correction 
in seconds of arc. 

For 51 Cephei, multiply the square of the time from elonga- 
tion in minutes by 0.124, and the product will be the correction 
in seconds of arc. 

The formula for reduction, when near elongation, is 

c = 1 12.5 f sin \" tan A, 

where c = correction to observed azimuth in seconds of arc ; 
t = time from elongation in seconds of time ; 
A = azimuth of star at elongation. 

log 1 12.5 sin 1" = 6.7367274. 

From this formula and that of equation (1) we may compute 
the coefficients for the above approximate rules for any latitude. 



5 18 SURVEYING. 



Thus, for latitude 30 we have azimuth of Polaris, 1885, l ° 
3C/.4, whence the coefficient of reduction for elongation of 
Polaris in latitude 30 is found to be 0.052, and for latitude 50 
it is 0.069. 

For 51 Cephei, this coefficient for latitude 30 is o.Iio, and 
for latitude 50 , 0.148. 
x From the above data the corrections for an observation of 
a circumpolar star near elongation may be computed. 

If azimuth be reckoned from the south point, as is common 
in topographical and other geodetic work, and if it increase in 
the direction S.W.N. E., then a star at western elongation has 
an azimuth of less than 180 , and at eastern elongation its 
azimuth is more than 180 . 

The corrections to reduce to elongation, as above com- 
puted, should be added to the computed azimuth of the star at 
western elongation, and subtracted when at eastern elongation. 

380. The Target. — This may be a sort of box, in which a 
light may be placed. A narrow vertical slit should be cut, sub- 
tending an angle, at the instrument, from one to two seconds of 
arc. This should be set as far from the instrument as conven- 
ient, as from a quarter of a mile to one mile. The width of 
slit desired may be computed for any given angular width 
and distance by remembering that the arc of one second is 
three-tenths of an inch for a mile radius. The target should 
be sufficiently distant to enable it to be seen with the stellar 
focus without appreciable parallax, as the instrument should 
not be refocused on the target. This target may be set on 
any convenient azimuth from the observation-station, as upon 
one triangulation station when the observations are taken at 
another, thus obtaining directly the azimuth of this line. 

381. Illumination of Cross-wires. — Various methods are 
used to illuminate the wires, the crudest of which is, perhaps, to 
hold a bull's-eye lantern so as to throw light down the tele- 
scope-tube through the objective, taking care not to obstruct 
the line of sight. 




GEODETIC SURVEYING. 519 

A very good reflector may be made from a piece of new 
tin, cut and bent as in Fig. 144. The straight 
strip is bent about the object end of the tele- 
scope tube, leaving the annular elliptic piece 
projecting over in front. This is then bent to 
any desired angle, preferably about forty-five 
degrees, and turned so that an attendant can FlG - 1 <4- 

reflect light down the tube by illuminating the disk from 
a convenient position. This position should be so chosen 
that the lantern may throw the light from the observer, 
rather than towards him. If the reflecting side of the disk be 
whitened, the effect is very good. The opening should be about 
three-fourths or seven-eighths inch in its shorter diameter, the 
longer diameter being such as to make its vertical projection 
equal to the shorter one. There is, of course, no necessity of 
limiting or of making true the outer edges of the disk. 



TIME AND LONGITUDE. 

382. Fundamental Relations. — In all astronomical compu- 
tations the observer is supposed to be situated at the centre 
of the celestial sphere and the stars appear projected upon its 
surface. Their positions with respect to the observer may be 
fixed by two angular coordinates. The most common plane of 
reference for these coordinates is that of the celestial equator, 
and the coordinates referring to it are known as Right Ascen- 
sion and Declination — corresponding to Longitude and Lati- 
tude on the earth's surface. * 

Right ascension is counted on the equator from west tow- 
ards east. As a zero of right ascension the vernal equinox 
is taken. 

Declination is counted on a great circle perpendicular to 
the equator, and is called positive when the star is north and 
negative when south. 



520 SURVEYING. 



In Fig. 145 

P is the pole ; 

Z is the zenith of the observer ; 

S is the star ; 

Then R. A. star = VPS = arc VE ; 

Dec. star = SE. 

These coordinates are fixed, varying only by slow changes 
due to the shifting of the reference-plane. 

Another system of coordinates is often used in fixing the 
place of a star, namely: Hour-angle and Declination. Hour- 
angle is the angle at the pole between the meridian and the 
great circle passing through the star and the pole perpendicu- 



Fig. 145. 

lar to the equator. Hour-angle will of course be constantly 
changing each instant. In Fig. 145 hour-angle = ZPS. 

383. Time. — The motion of the earth on its axis is perfect- 
ly uniform. We obtain, therefore, a uniform measure of time 
by employing the successive transits of a point in the equator 
across the meridian of any place. The point naturally chosen 
is the vernal equinox. 

A Sidereal Day is the interval of time between two succes- 



GEODETIC SURVEYING. 521 

sive upper transits of the vernal equinox over the same merid- 
ian. 

The Sidereal Time at any instant is the hour-angle of the 
vernal equinox at that instant reckoned from the meridian 
westward from o h to 24 11 . Thus, when the vernal equinox is on 
the meridian, the hour-angle is o h o m s and the sidereal time 
is o h o m o s . When the vernal equinox is i h west of the merid- 
ian the sidereal time is i h o m O a . 

We have in Fig. 145 

Hour-angle of ver. eq. = ZPV = 6 = sidereal time ; 
Right asc. of star == VPS == a ; 

Hour-angle of star = ZPS — H\ 

Whence 6 - a = H. (1) 

From this equation, knowing the sidereal time and the 
R. A. of the star, the hour-angle may always be computed. 

When H = o, i.e., when the star is on the meridian, 6 — a f or, 
in other words, the R. A. of any star is equal to the true local 
sidereal time when the star is on the meridian. By noting the 
exact time of transit of any star whose R. A. is known, the 
local sidereal time will be at once known. 

An Apparent Solar Day is, the interval of time between two 
successive upper transits of the true sun across the same 
meridian. 

Apparent or True Solar Time is the hour-angle of the true 
sun. 

Owing to the annual revolution of the earth, the sun's 
right ascension is constantly increasing. It follows, therefore, 
that a solar day is longer than a sidereal day. In one year 
the sun moves through 24 brs of right ascension. There will 
be, therefore, in one tropical year (which is the interval be- 



522 SURVEYING. 



tween two- successive passages of the sun through the vernal 
equinox) exactly one more sidereal day than solar days ; or, in 
other words, in a tropical year the vernal equinox will cross 
the meridian of any given place once more than the sun will. 

The solar days will, however, be unequal for two reasons : 

1st. The sun in its apparent motion round the earth does 
not move in the equator, but in the ecliptic. 

2d. Its motion in the ecliptic is not uniform. 

On account of these inequalities the true solar day cannot 
be used as a convenient measure of time. But a mean solar 
day has been introduced, which is the mean of all the true 
solar days of the year and which is a uniform measure of 
time. 

Suppose a fictitious sun to start out from perigee with the 
true sun, to move uniformly in the ecliptic, returning to peri- 
gee at the same moment as the true sun. Now, suppose a 
second fictitious sun moving in the equator in such a way as 
to make the circuit of the equator in the same time that the 
first fictitious sun makes the circuit of the ecliptic, the two fic- 
titious suns starting together from the vernal equinox and re- 
turning to it at the same moment. The second fictitious sun 
will move uniformly in the equator and will be therefore a 
uniform measure of time. This second fictitious sun is known 
as the Mean Sun. 

A Mean Solar Day is therefore the interval between the 
upper transits of the mean sun over the meridian of any place. 

Mean Solar Time at any meridian is the hour-angle of the 
mean sun at that meridian counted from the meridian west 
from o h to 24 hrs . 

The Equation of Time is the quantity to be added to or 
subtracted from apparent solar time to obtain mean time. 

The equation of time is given in the American Ephemeris 
for Washington mean and apparent noon of each day. If the 
value is required for any other time it can be interpolated be- 
tween the values there given. 



GEODETIC SURVEYING. $2$ 

384. To convert a Sidereal into a Mean-time Interval, 
and vice versa. — According to Bessel, the tropical year con- 
tains 365.24222 mean solar days, and since the number of side- 
real days will be greater by one than the number of mean solar 
days, we have 

365.24222 mean sol. days = 366.24222 sid. days ; 
I mean sol. day = 1.0x3273791 sid. days. 

Let I m = mean solar interval ; 

I s = sidereal interval ; 
k = 1. 0027379 1. 

Thus 

I s = I m k = f m +.I m (k — 1) =I m + 0.00273794 ; 



By the use of these formulae the process of converting a 
sidereal interval into a mean-time interval, and vice versa, is 
made very easy. It is rendered more easy by the use of 
Tables II. and III. of the Appendix to the American Ephem- 
eris and Nautical Almanac, where the quantity I m (k—i) is 

given with the argument I m , and I 8 i 1 — — ] with the argument I s . 

Example. — Given the sidereal interval I s = 1 5 h 40 m 50 s . 50, find 
the corresponding mean-time interval. 



Im — T = J s — ■*• I 1 — T/ = f s ~ 0.0027304/ s . 





Is = 


I5 b 40 m 50 s . 50 


Table II. gives for 15 11 40 m 




2 33-99 6 


" " " 50 s .50 




0.138 



Im= 15 38 16.37 



524 . SURVEYING. 



385. To change Mean Time into Sidereal. — Referring 
to Fig. 145, suppose 5 to represent the mean sun. 

Then ZPS = hour-angle of mean sun = mean-time = T; 
VPE = R. A. of mean sun = a s ; 

6 = sidereal time. 
From equation (1), p. 521, 

d=a 8 +T. 

The right ascension of the mean sun is given in the Ameri- 
can Ephemeris both for Greenwich and Washington mean 
noon of each date. It is called ordinarily the sidereal time of 
mean noon, which is of course the right ascension of the mean 
sun at noon, since at mean noon the mean sun is on the 
meridian and its right ascension is equal to the sidereal time. 
Since the sun's right ascension increases 360 or 24 hr8 in one 
year, it will change at the rate of 3 m 56 s .555 in one day, or 
C/.8565 in one hour. 

Suppose 0/ = sid. time of mean noon at Greenwich ; 

6 = " " " " " " the place for which 

T is known ; 
L = longitude west of Greenwich. 
Then 6 Q = 8: + C/.8565Z, 

where L is expressed in hours and decimals of an hour. 
In this way the sidereal time of mean noon may be obtained 
for the meridian of observation. 

Substituting for a s its equivalent, and reducing the mean- 
time interval to sidereal, 

( = o + T+T(k-i). 

Example. — Longitude of St. Louis, 6 h o m 49 s . 16 = 6 h .oi36. 
Mean time, 1886, June 10, io h 25™ 25 s .5. Required correspond- 
ing sidereal time. 



GEODETIC SURVEYING. 525 

From Amer. Ephem., p. 93 : 

8 ' (for Greenwich) = $ h i$ m 3 s . 30 
6.0136 X 9- 8 5 6 5 = o 59.27 



/ 



T 




= 5 16 2.57 
= 10 25 25.50 


T\k 


?- 1), Table III., 


= 1 42.74 



6 — 15 43 10.81 

It should be remarked that the quantity 59 8 .27 will be a 
constant correction, to be added to the sid. time of mean noon 
at Greenwich to obtain the sid. time of mean noon at St. 
Louis. 

386. To change from Sidereal to Mean Time. — This 
process is simply the reverse of that for changing from mean 
to sidereal time. Using the same notation as before, we shall 
have 



T=0-6,-(9-6){i-l>). 



Subtracting from the given sidereal time (#) the sidereal 
time of mean noon (0 O ), we have the sidereal interval elapsed 
since mean noon, and this needs simply to be changed into a 
mean-time interval. 

Example. — Given 1886, June 10, 15 11 43 111 io s .8i sidereal 
time, to find the corresponding mean time. 

= 15 43 10.81 
(as before) O = 516 2.57 

— 6 = 10 27 8.24 

(0- o )(l-j) (Table II.) = x 42 . 74 

T= 10 25 25.50 



526 SURVEYING. 



387. The Observation for Time, as here described,* 
consists in observing the passage, or transit, of a star across 
the meridian. The direction of the meridian is supposed to 
have been determined by an azimuth observation. If the in- 
strument be mounted over a station the azimuth from which 
to some other visible point is known, the telescope can be put 
in the plane of the meridian. An observation of the passage 
of a star across the meridian will then give the local time, when 
the mean local time of transit of that star has been computed. 
In order to eliminate the instrumental errors at least tv/o stars 
should be observed, at about the same altitude. If the instru- 
ment has no prismatic eye-piece, then only south stars can be 
observed with the ordinary field-transits; that is, only stars 
having a south declination, if the observer is in about 40 north 
latitude. Stars near the pole should not be chosen, since they 
move so slowly that a small error in the instrument would 
make a very large error in the time of passage. 

388. Selection of Stars. — The stars should be chosen in 
pairs, each pair being at about the same altitude, or declination. 
It is supposed that the American Ephemeris is to be used. 
The " sidereal time of transit, or right ascension of the mean 
sun," is its angle reckoned easterly on the equatorial from the 
vernal equinox. This is given in the Ephemeris for every day 
of the year. Similarly, the right ascension of many fixed stars 
is given for every ten days of the year, under the head of 
" Fixed Stars, Apparent Places for the Upper Transit at Wash- 
ington." These latter change by a few seconds a year, from 
the fact that the origin of coordinates, the vernal equinox itself, 
changes by a small amount annually. If, therefore, the hour- 
angle, or right ascension, of both the mean sun and a fixed 



* It is assumed that the engineer or surveyor has only the ordinary field- 
transit, without prismatic eye-piece, so that he can only read altitudes less than 
6o°. The accuracy to be attained is about to the nearest second of time. 



GEODETIC SURVEYING. 527 



star be found for any day of the year, the difference will be the 
sidereal interval intervening between their meridian passages, 
the one having the greater hour-angle crossing the meridian 
much later than the other. When this interval is changed 
to mean time the result is the mean or clock time intervening 
between their meridian passages. If a fixed star is chosen 
whose right ascension is eight hours greater than that of the 
mean sun for any day in the year, then this star will come 
te the meridian eight hours (sidereal time) after noon, or at 
7 h 58 m 41 s . 364 after noon of the civil day indicated in the Nau- 
tical Almanac. If, therefore, one wishes to make his observa- 
tions for time from 8 to 10 o'clock P.M. he should select stars 
whose hour-angles, or right ascensions, are from 8 to 10 hours 
greater than that of the mean sun for the given date. 

In the following table such lists are made out for the first day 
of each month for the year 1888. The mean time of transit is 
given for the meridian of Washington to the nearest minute, as 
well as its mean place for the year. None of these values will 
vary more than three or four minutes from year to year, and 
therefore the table may be used for any place and for any time. 
The table merely enables the observer to select the stars to be 
observed. After these are chosen their local mean time of transit 
must be worked out with accuracy from the Nautical Almanac* 
For any other day of the month we have only to remember that 
the star comes to the meridian 3 m 56 s earlier (mean time) 
each succeeding day, so that for n days after the first of the 
month we subtract 3.93 n minutes from the mean time of 
transit given in the table, and this will give the approximate 
mean time of transit for that date. If we take n days before 

* Even this trouble may be avoided by using Clarke's Transit Tables (Spon, 
London). Price to American purchasers less than one dollar. They are pub- 
lished annually in advance, and give the Greenwich mean time of transit of the 
sun and many fixed stars for every day in the year. They are computed for pop- 
ular use from the Nautical Almanac. 



528 



SURVEYING. 



H 
O 



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w 

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o 
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in in m >n in vo 



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GEODETIC SURVEYING. 



529 



s 

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8 

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m to m o w f> o 
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34 



530 SURVEYING. 



a date in the table, add 2.93 n minutes to the corresponding 
time of transit to find the approximate time of transit for the 
given date. This table is therefore a mere matter of conve- 
nience to assist in selecting the stars to be used. They are 
nearly all southern stars, since these only can be observed with 
the ordinary field-transit. 

389. Finding the Mean Time of Transit. — As explained 
above, the mean or clock time of transit is simply the sidereal 
interval between the mean sun and star for the given place and 
date, reduced to mean time. To find this interval we find the 
right ascension of both mean sun and star, and take their dif- 
ference. But the right ascension or sidereal time of the mean 
sun or mean noon is given for the meridian of Greenwich, 
whereas by the time the sun has reached the given American 
meridian its right ascension or sidereal time has increased 
somewhat, the hourly increase being 9 S .8565. To find the 
" sidereal time of mean noon" for the given place, therefore, 
we take the value for the given date for Greenwich and add 
to it 9 S .8565 for every hour of longitude the place is west of 
Greenwich. This then gives the "local sidereal time of mean 
noon." The right ascension of the star, or the sidereal time of 
its meridian passage, is then found. This changes only by a 
few seconds in a year, and is given for every ten days in the 
Washington Ephemeris. This, therefore, needs no correction 
to reduce it to its local value for any place. The difference 
between the " local sidereal time of mean noon" and the sidereal 
time of the star is the sidereal interval of time elapsing between 
local mean noon and the transit of the star. When this sidereal 
interval is changed to a mean-time interval, which is effected 
by means of a table at the back of the Nautical Almanac, the 
result is the local mean time of transit of the star. 



GEODETIC SURVEYING. 



531 



Example. — Compute the local mean time of transit of e Eridani at St. Louis 
on Jan. 16, 1888. 

Sidereal time of mean noon at Greenwich = I9 h 41™ 27 s . 80 
Correction for longitude 6.05 11 west 

Local sidereal time of mean noon 
Right ascension e Eridani Jan. 16 

Sidereal interval after mean noon 
Correction to reduce to mean time 

Local mean time of transit 



= 


+ 




59 


.63 


= 


19 


42 


27 


•43 


^z 


3 


27 


39 


.21 


= 


7 


45 


11 


•78 


■=. 


— 


1 


16 


.21 



= T 43 m 55 8 .57 



// 




390. Finding the Altitude. — The relation between lati- 
tude, declination, and altitude is shown by Fig. 146, which rep- 
resents a meridian section of the celestial 
sphere. Let PP be the line through the 
earth's axis ; QQ the plane of the equa- 
tor; Z the zenith, and HH' the horizon. 
Then H'P=ZQ = is the latitude of 
the place, and QS = 6 and QS" = —6 
are the declinations of 5 and S" respec 
tively. The altitude of the star 5 is H'S y Fig. 146. 

or measured from the south point it would be HS. The alti- 
tude of the star S" is US". 

We have therefore for altitude of »S 

f = HZ- ZQ+ QS = 90 - 0+ 8. 

Also for altitude of S'\ 

h" = HZ- ZQ- QS" = 90 - - 6" 

But since south declination is considered as negative, we 
have, in general, for altitude from the south point, of a star in 
the meridian, 

A = 90 - + 6. 



The latitude is supposed to be known and the declination 



53 2 SURVEYING. 



is given in the table, whence the altitude of any star in the 
list is readily found. 

391. Making the Observations. — The meridian is sup- 
posed to be established. This may be done either by having 
two points in it fixed, one of which is occupied by the instru- 
ment and the other by a target, or an azimuth may be known 
to any other station or target. In either case the instrument 
is put into the meridian by means of both verniers, either mak- 
ing the mean of the two read zero on the meridian post, or by 
making the mean of their readings on the azimuth station dif- 
fer from their mean reading in the meridian by an amount 
equal to the azimuth of the given line. 

Or, the setting may be approximately on the meridian and 
the angle measured so that the true deviation of the instru- 
ment from the meridian is observed for each star observation. 
The error in time, from a given small error in azimuth, is then 
found from the differential equation* 

# = *"«-*> da, 
cos o 

where dt is the error in hour-angle in seconds of arc when da 

is the deviation from the meridian in seconds of arc, $ being 
the latitude of the place, and 6 the declination of the star. 

, . . * — ■ ™" 

* One of the fundamental equations that may be written from an inspection 
of Fig. ii, p. 49, is 

cos 8 sin t = — cos h sin a, 
where h is the altitude and t, 8, and a as above. Differentiating with reference 

to t and «, we have 

cos h cos a . 

dt = x - da. 

cos o cos t 

For observations very near the meridian both cos a and cos t become 
unity, and then we have 

cos o cos o 



GEODETIC SURVEYING. 533 

Having found the time correction in seconds of arc, the 
correction in seconds of time is found by dividing by fifteen. 

If the declination is south, or negative, the equation be- 
comes 

cos o 



The error from this cause diminishes as the altitude of the 
star increases, and is zero for a zenith observation. 

The stars are chosen in pairs, the two stars of a pair hav- 
ing about the same altitude or declination. Thus, from the 
January group we might select o 1 Eridani and § Ononis as 
one pair, and fi Eridani and r Orionis for. another. The stars 
are of course observed in the order of their coming to the 
meridian, irrespective of the way they are paired, but they are 
paired in the reduction. 

The visual angle of the field of view of the ordinary engi- 
neer's field-transit is' something over one degree. The star 
will therefore be visible in the telescope for something over 
two minutes before it comes to the vertical wire, it being here 
assumed that there is but one vertical thread. Let an attend- 
ant hold the watch or chronometer and note the time to the 
nearest second when the star is on the wire, as noted by the 
observer. If this time be compared with that of the computed 
mean time of transit, the error of the chronometer is obtained, 
so far as this observation gives it. 

The instrument must be reversed on the second star of each 
pair. This is to eliminate the instrumental errors. The hori- 
zontal angle to the station-mark (whether this be on the 
meridian or not) should also be read for every reading on a 
star, or at least before and after the star-readings. 

The following programme would be adapted to observa- 
tions on the four stars selected above : 



534 SURVEYING. 



PROGRAMME. 

i. Set on azimuth station and read horizontal angle (both 
verniers). 

2. Set in the meridian and read both verniers. 

3. Set the approximate altitude of o 1 Eridani. 

4. Note time of passage of o 1 Eridani. 

5. Set on azimuth station and read both verniers. 

6. Set in the meridian and read verniers. 

7. Note time of passage of J3 Eridani. 

8. Revolve the telescope 180 on its horizontal axis, relevel, 
and read on the azimuth station. 

9. Set in the meridian and read verniers. 

10. Note time of passage of /3 Ononis. 

11. Note time of passage of r Ononis. 

12. Read both verniers again in the meridian before the 
instrument is disturbed. 

13. Read to azimuth station. 

We have thus obtained four measurements of the hori- 
zontal angle, and read with the telescope normal and inverted 
on each pair of stars. Especial care must be takeji to see that 
the plate-bubble set perpendicular to the telescope is exactly 
in the centre when readings are taken to the stars. The mean 
chronometer error for the two stars of a pair is its true error, 
provided it has no rate. If the chronometer has a known rate, 
that is, if it is known to be gaining or losing at a certain rate, 
then its error must be found for some particular time, as that 
of the first observation. Its rate must then be applied to the 
observed time of transit of the other stars for the intervening 
intervals before comparing results. If local time alone is de- 
sired, the result is obtained as soon as a pair of stars has been 
observed and their mean result found. 

392. Longitude. — If geographical position or longitude is 
sought, it remains to compare the chronometer with the 



GEODETIC SURVEYING. 535 

standard or meridian time for that region. This standard time 
is now transmitted daily from fixed observatories to almost all 
railroad stations in the United States. The time thus trans- 
mitted is probably never in error more than a few tenths of a 
second. It is usually sent out from io A.M. to noon daily. If 
the rate of the station clock is known, and also that of the 
watch used in the time observation, then a comparison of these 
subsequent to the observation would give the difference be- 
tween local time and the hourly meridian time used, which 
difference changed to longitude would be the longitude of the 
place east or west of that standard meridian. If the station 
clock cannot be relied on as to its rate, then the watch used in 
the observation must have a constant known rate. In this 
case the observer compares his watch on the following day 
with the time signal as it is transmitted over the railroad com- 
pany's wires, and so obtains his longitude. 

Local time can be observed in this way by means of an ordi- 
nary transit to the nearest second of time, and the longitude ob- 
tained to the same accuracy if the rate of the chronometer used 
is constant and accurately known. It is probable, however, 
that several seconds error may be made if a watch is used, 
since probably no watch has a rate which is constant within 
one second in twelve hours. Therefore if longitude is desired 
a portable chronometer should be used whose rate is well 
known.* 

393. Computing the Geodetic Positions. — After the 
angles of the system are adjusted, and the sides of the triangles 
computed, we have the plane angles and linear distances from 
point to point in the system. It now remains to compute the 



*This method has been extensively used for obtaining approximate geodetic 
positions for the U. S. Geological Survey in the West, comparisons being made 
daily with the Washington University time signals which are transmitted to the 
railways in that region. 



53$ SURVEYING, 



latitudes and longitudes of the several stations, and the azi- 
muths of the lines. 

The following formulae, though not exact, are quite suffi- 
cient when the sides of the triangles do not exceed ten or 
fifteen miles in length : * 

NOTATION. 

Let L! = latitude of the known point ; 
L — latitude of the unknown point; 
M' = longitude of the known point; 
M = longitude of the unknown point ; 
Z' = azimuth of the unknown point from the known, 

counting from the south point in the direction 

S.W.N. E.; 
Z '= azimuth of the known point from the unknown, 

or the back azimuth ; 
K = length in metres of line joining the two points; 
e = eccentricity of the earth's meridian section; 
N = length of the normal, or radius of curvature of a 

section perpendicular to the meridian of the 

middle latitude, in metres. 
R = radius of curvature of the meridian in metres. 

Then we have in terms of the length and azimuth of a 
given line, in seconds of arc, when the distances are given in 
metres, 

V -L= -dL = BK cos Z+ CK* sin 2 Z + DA* n 

M'-M= +dM=AK^— F ; Yd) 

1 cos L ^ > 

Z , + iSo°-Z= - dZ=dM sin L m ; J 

* For a summarized derivation of these formulae for computing the L M 
Z's from triangulation data, together with extended tables of factors used, see 
Report U. S. Coast and Geodetic Survey, 1884, Appendix No. 7. The deriva- 
tion of the formula? is further amplified in Appendix D of this book. 



GEODETIC SURVEYING. 



537 



where A* = 



Narc I' 



B = 



Rare I 



tan L 



2RN*XQ.\'" 



D 



fV sin L cos L sin i" 



mean latitude = 



L + L' 



(I - e* sin 2 Z)f ' w 2 

and A = value of first term in right member = BK cos Z. 
Careful attention must be paid to the signs of the Z functions. 



TABLE OY LM Z COEFFICIENTS. 



Latitude. 


Log. A -\- 10. 


Log. B -\- 10. 


Log. C + 10. 


Log. D + 10. 


30° 


8.5093588 


8.5115729 


I. 16692 


2.3298 


31 


3363 


5054 


.18416 


•3382 


32 


3134 


4368 


.20I08 


.3460 


33 


29OI 


3669 


.21772 


•3532 


34 


2665 


2959 


. 23409 


•3597 


35 


8.5092425 


8.5112239 


I.25024 


2.3656 


36 


2182 


I5IO 


.26617 


.3709 


37 


I936 


0772 


.28193 


•3756 


38 


1687 


8. 5 1 10027 


•29753 


•3797 


. 39- 


1437 


8.5109275 


.31299 


.3833 


40 


8.5091184 


8.5108517 


I.32833 


2.3863 


4i 


O930 


7755 


.34358 


.3888 


42 


0675 


6989 


.35875 


•39°7 


43 


0419 


6220 


•37386 


.3921 


44 


8.5090162 


5449 


.38894 


• 3930 


45 


8.5089904 


8.5104677 


I . 40400 


2-3933 


46 


9647 


39°5 


.41906 


•3932 


47 


9390 


3134 


.43414 


.3924 


48 


9133 


2364 


.44926 


.3912 


49 


8878 


1598 


.46443 


.3894* 


50 


8.5088623 


8.5100835 


I.47968 


2.3871 



* A is to be evaluated for L. 

\ This denominator is given to the f power in the Coast Survey formulae. 
The rigid development used in Appendix D. shows it to be as given above. 
The error, however, is small. 



53^ SURVEYING. 



Logarithmic values of the coefficients A, B, C, and D are 
given in the above table for each degree of latitude from 
30 to 50 . By the aid of this table the LMZ's are readily 
found. These tabular values are computed from the constants 
of Clarke's spheroid. In this we have 

Equatorial semidiameter = 6 378 206 metres. 
Polar semidiameter = 6 356 584 " 

Whence the ratio of the semidiameters is - — . 

293.98 

Clarke's value of the metre has been taken, which is 
1 metre = 39.37043 inches. 

The difference of azimuth of the two ends of a line is due 
to the convergence of the meridians passing through its ex- 
tremities, this convergence, as seen from the last of equations 
(1), being equal to the difference of longitude into the sine of 
the mean latitude. 

When the sides of a system of triangulation have been 
computed, and the azimuths of the lines are desired from the 
several stations, the successive differences of latitude and 
longitude are first computed, and from these the azimuths of 
the lines, using equations (1). If the longitude is unknown, 
the longitude of the first station may be assumed without 
affecting the accuracy of the computed relative positions. The 
last of equations (1) gives the difference between the forward 
and ba*ck azimuth of the line joining the two stations. This 
difference being applied, with the proper sign, gives the azi- 
muth of the first station as seen from the second. But when 
the azimuth of one line from a station is known, the azimuths 
of all other lines from that station are found from the adjusted 
plane-angles at that station, provided the spherical excess had 



GEODETIC SURVEYING. 



539 



been deducted or allowed for, in the adjustment. If no ac- 
count has been taken of spherical excess, the error in azimuth 
accumulates in working eastward or westward, and soon be- 
comes appreciable. 

For any other station the azimuth correction is again found 
for the line joining this station with a station where azimuths 
have been computed, which when applied gives the azimuth of 
this line as taken at the forward station, whence the azimuths 
of all the lines from this station are known, and so on. 



394. Example. — In Fig. 142, p. 495, let the azimuth of the line CA, from 
C, be 8o°; latitude of Cbe 40 ; the length of the line CD be 25000 metres (over 
15 miles) ; required the geodetic position of D, and the azimuth of the line DC 
from D. 

COMPUTATION OF L M Z. 



Z> 
dZ 

180 
Z 



C to A 

A CD = C b (see p. 495) 

C toZ> 

D to C 



80 8 
39 



119 



180 
299 



38 



00". o 
06 .1 



06 .1 
49 -5 



16 .6 



L' 
dL 



40° 
+ 


00' 

6 


00". 000 
41 .8*7 


40 


06 


41 .847 



c 

25000 metres. 
D 



M' 
dM 


90 

+ 


oo' 
15 


M 


90 


15 



00 .000 
16 .019 



16 .019 



1st terra 
2d and 3d terms 

- dL 



-402". 853 
-f- 1 .006 



— 401 .847 



40" 03' 22' 



B 

K 

cos Z 



A 

K 

sin Z 

cos L (a.c.) 

dM 



8.5108517 


C 


4.3979400 


K* 


9.6963560 


sin 2 Z 


2.6051477 


8. 5091 156 




4.3979400 




9.9383948 


dM 


0.1164540 


sinZ m 


2.9619044 




4- 9i6".oi9 


-dZ 



1.32833 
8.79588 
9.87679 


D 


O.OOTOO 
I.OO23 


2.96190 
9.80857 

2.77047 
+589". 48 





2.3863 
5.2103 



7.5966 
0.00395 



540 



SURVEYING. 



GEODETIC LEVELLING. 

395. Geodetic Levelling is of two kinds : (A) Trigonomet- 
rical Levelling and (B) Precise Spirit-levelling. In trigonomet- 
rical levelling the relative elevations of the triangulation-sta= 
tions are determined by reading the vertical angles between the 
stations. When these are corrected for curvature of the earth's 
surface and for refraction it enables the actual difference of 
elevation to be found. In precise spirit-levelling a special type 
of the ordinary spirit or engineer's level is used, and great 
care taken in the running of a line of levels from the sea-coast 
inland, connecting directly or indirectly with the triangulation 
stations and base-lines. Both these methods will be described. 

(A) TRIGONOMETRICAL LEVELLING. 

396. Refraction. — If rays of light passed through the atmos- 
phere in straight lines, then in trigonometrical levelling we should 




have to correct only for the curvature of a level surface at the 
locality. It is found, however, that rays of light near the sur- 
face of the earth usually are curved downwards — that is, their 
paths are convex upwards. This curve is quite variable, some- 
times being actually convex downwards in some localities. It 



GEODETIC SURVEYING. 541 

has its greatest curvature about daybreak, diminishes rapidly 
till 8 A.M., and is nearly constant from 10 A.M. till 4 P.M., when 
it begins to increase again. The curve may be considered a 
circle having a variable radius, the mean value of which is 
about seven times the radius of the earth. 

397. Formulae for Reciprocal Observations. — In Fig. 147 
the dotted curve represents a sea-level surface. 

Let H = height of station B above sea-level ; 
H' == height of station A above sea-level ; 
C = angle subtended by the radii through A and B ; 
2T= true zenith distance of A as seen from B; 
Z = true zenith distance of B as seen from A ; 
d = true altitude of A as seen from B = go° — Z ; 
6' = true altitude of B as seen from A = 90 — Z' ; 
h = apparent altitude of A as seen from B = 6 -|- re- 
fraction ; 
h! = apparent altitude of B as seen from A = d' -f- re- 
fraction. 
d= distance at sea-level between A and B; 
r = radius of the earth ; 
m = coefficient of refraction. 

In the figure join the points A and B by a straight line. 
This would be the line of sight from A to B if there were no 
refraction. Through A and B draw the radii meeting at C, ex- 
tending them beyond the surface.* Take the middle point of 
the line AB, as H, and draw HC. Take A A ' perpendicular to 
HC, and EE' through H and perpendicular to HC. Extend 
AA r to meet a perpendicular to it from B. Then do we have 

A'C = AC: E'E = AD; and HC = r + HxH '' 

2 

* In reality these are the normals at A and B, but will here be taken as the 
radii. 



54 2 SURVEYING. 



The last relation is exact ; the first two are not exact, be- 
cause HC does not quite bisect the angle C. The figure is 
greatly exaggerated as compared to any possible case in prac- 
tice, for the angle C would never be more than i° in such work. 
The error in practice is inappreciable. 

From the geometrical relations shown in the figure we 
have 

H - IT = A'B=DB sec- (i) 

2 v ' 

But since Cis never more than i°, and usually much less, 
we may say 

H-H f =A f B = DB = ADtznBAD.. . . (2) 

But AD = E r E= distance between the stations reduced to 
their mean elevation above sea-level = d' ; also 

BAD = \(Z - Z') ; 
/. H-H' = d't3.ni(Z-Z'). .... (3) 

But since d = distance between stations at sea-level, we 
have 

a : d : : r-\ : r, 

1 2 ' 

*=-/(i+^±^); (4j 

whence we have, for reciprocal observations at A and B, 

ff~H> = Jtani{Z-Z')(i+¥±¥), . . (5) 



GEODETIC SURVEYING. 543 

or, in terms of d and & ', 

ff-ff^tanF-^'l-3 • • (6) 

where attention must be paid to the signs of # and S\ 

The effect of refraction is to increase d and d' by equal 
amounts (presumably), whence their difference remains unaf- 
fected. Equations (5) and (6) are therefore the true equations 
to use for reciprocal observations at two stations. Since the 
refraction is so largely dependent on the state of the atmos- 
phere, the observations should be made simultaneously for the 
best results. This is seldom practicable, however, and therefore 
it is highly probable that a material error is made in assuming 
that the refraction is the same at the two stations when the 
observations are made at different times. 

398. Formulae for Observations at One Station only. — 
If the vertical angle be read at only one of the two stations, 
then the refraction becomes a function in the problem. Since 
the curve of the refracted ray is assumed to be circular (it 
probably is not when stations have widely different elevations), 
the amount of angular curvature on a given line is directly pro- 
portional to the length of the line or to the angle C. The dif- 
ference at A or B between the directions of the right line AB 
and the ray of light passing between them is one half the 
total angular curvature of the ray ; that is, it is the angle 
between the tangent to the curved ray at A and the cord AB. 
The ratio between this refraction angle at A or B and the 
angle C is a constant for any given refraction curve ; that is, 
this ratio does not change for different distances between sta- 
tions. This ratio is called the coefficient of refraction, and is 

C 
here denoted by m. The true angle BAD is equal to 6' -| — , 

but since the observed altitude is increased by the amount of 



544 SURVEYING. 



the refraction, we have for the apparent altitude of B, as seen 
from A, 

k' = S f + mC\ 

C 

whence BAD — k'-\--— mC. • (7) 

Using this value of the angle BAD in equation (2), we 
obtain 

R~H , = d / tan [h! + ^- mC) 

-^tan(^ + f-;^)(i+^±;-), . (8) 

where h! is positive above and negative below the horizon. 

Equation (8) is used where the vertical angle is read from 
one station only. 

Since the total angular curvature of the ray of light between 
A and B is 2ntC, and the curvature of the earth is C, we may 
write 

v 
C: 2mC :: r' : r, or r' = — , ... (9) 

where r' is the radius of the curve of the refracted ray. 

Since the curvature of the ray is of the same kind as that 
of the earth, but less in amount, the total correction for curva- 

C C 

ture and refraction is for an angle equal to mC^= — (1 — 2m). 

Also, since C is always a small angle, we may put 

C (in seconds of arc) == — : 7l . 

J r sin 1" 

If the mean radius is used, we have, in feet, 

log r — 7.32020, and log sin 1" = 4.6855749, 



GEODETIC SURVEYING. 



545 



whence in seconds of arc and distance in feet we have 



or 



log C = log d — 2.00577 

101.34 



(10) 



or the curvature is approximately equal to 1" for 100 feet in 
distance. 

The following table gives computed values of the combined 
mean corrections for curvature and refraction for short dis- 
tances, either for horizontal or inclined sights. Both the dis- 
tance d and the correction c n are in feet, except for the last 
column, where the distance is given in miles. For a more ex- 
tended table for long distances, see page 433. 





CORRECTION FOR EARTH'S CURVATURE 


AND REFRACTION. 




d 


c n 


d 


C a 


d 


C n 


d 


C n 


d 


C n 


miles. 


C n 


300 


.002 


1 13 00 


•°35 


2300 




108 


3300 


.223 


4300 


379 


1 


•571 


400 


.003 


1400 


.040 


2400 




118 


3400 


•237 


4400 


397 


2 


2.285 


500 


.005 


! 1500 


.046 


2500 




128 


3500 


.251 


4500 


415 


3 


5-142 


600 


.007 


1600 


.052 


2600 




139 


3600 


.266 


4600 


434 


4 


9. 141 


700 


.010 


1700 


•059 


2700 




149 


3700 


.281 


4700 


453 


5 


14.282 


800 


.013 


1800 


.066 


2800 




161 


3800 


.296 


4800 


472 


6 


20.567 


900 


.017 


1900 


.074 


2900 




172 


3900 


.312 


4900 


492 


7 


27.994 


1000 


.020 


2000 


.082 


3000 




184 


4000 


.328 


5000 


512 


8 


36-563 


1 100 


.025 


2100 


.090 


3100 




197 


4100 


•345 


5100 


533 


9 


46.275 


1200 


.030 


2200 


.099 


3200 




210 


4200 


.362 


5200 


554 


10 


57-13° 



399. Formulae for an observed Angle of Depression to 
a Sea Horizon. — In Fig. 148 let A be the point of observa- 
tion and 5 the point on the sea-level surface where the tangent 
from A falls. Then we have 

H=AI)=*ASt*xiASD 

C 
=rrtan£Ttan- (n) 



* Let the student prove this relation. 



35 



546 



SURVEYING. 



Since the angle C is always very small, we may let the arc 
A r*^ k. — : ■ equal its tangent, whence 




H = - tan 2 C . . (12) 
2 v J 



If the observed angle of de- 
pression be h = C — mC y 



then 

and 

or 



C = 



h 



1 — n£ 



2 



\i — mr 



H = ^(-A-Y tan' I", 

2\i — ml 



where h is expressed in seconds of arc. 

Log - tan 3 \" = 6.39032 for distances in feet. 
400. To find the Value of m we have 



whence 



or 



or 



Z = 90 - h + mC, 
Z + Z'= i8o° + C = 1S0 - k - k' + 2mC, 



1 — 2m = 



m 



C ' 



(13) 
(14) 



(1) 



GEODETIC SURVEYING. $47 



where h and h! are the observed altitudes above the horizon. 
It is evident that every pair of reciprocal observations at two 
stations will give a value for m. The mean values of m y as 
found from observations on the United States Coast Survey 
in New England, were : 

Between primary stations, . . . . m = 0.071 

For small elevations, m = 0.075 

For a sea horizon, m = 0.078 

■ 

On the New York State Survey the value from 137 obser- 
vations was m = 0.073.* 



MA- M' 
In this work also the term in equations (4) to (! 

never affected the result by more than 3-0V0 P art °f * ts value. 



PRECISE SPIRIT-LEVELLING. 

401. Precise Levelling differs from ordinary spirit-level- 
ling both in the character of the instruments used and in the 
methods of observation and reduction. It is differential 
levelling over long lines, the elevations usually being referred 
to mean sea-level. In order that the elevations of inland 
points, a thousand miles or more from the coast, may be de- 
termined with accuracy, the greatest care is required to pre- 
vent the accumulation of errors. In order that triangulation 
distances may be reduced to sea-level, the elevations of the 
bases at least must be found. It is impossible to carry eleva- 
tions accurately from one triangulation-station to another by 
means of the vertical angles, on account of the great variations 
in the refraction. Barometric determinations of heights are 
also subject to great uncertainties unless observations be 

* See pages 435 and 436 for a case of excessive refraction profitably 
utilized. 



;4§ * SURVEYING. 



made for long periods. The only accurate method of finding 
the elevations of points in the interior above sea-level is by- 
first finding what mean sea-level is at a given point by means.of 
automatic tide-gauge records for several years, and then run- 
ning a line of precise spirit-levels from this gauge inland and 
connecting with the points whose elevations are required. 
Most European countries have inaugurated such systems of 
geodetic levelling, this work being considered an integral part 
of the trigonometrical survey of those countries. In the 
United States this grade of work was begun on the U. S. feake 
Survey in 1875, by carrying a duplicate line of levels from 
Albany, N. Y., and connecting with each of the Great Lakes. 
The Mississippi River Commission has carried such a line 
from Biloxi, on the Gulf of Mexico, to Savannah, 111., along 
the Mississippi River, and thence across to Chicago, connect- 
ing there with the Lake Survey Elevations * The U. S. Coast 
and Geodetic Survey is carrying a line of precise f levels from 
Sandy Hook, N. J., across the continent, passing through St. 
Louis, their line here crossing that of the Mississippi River 
Commission. On all these lines permanent bench-marks are 
left at intervals of from one to five miles, whose elevations 
above mean sea-level are determined and published. 

402. The Instruments used in precise levelling differ in 
many respects from the ordinary wye levels used in America. 
The levelling instrument prescribed by the International Geo- 
detic Commission held in Berlin in 1864 is shown in Fig. 149. 

These instruments are made by Kern & Co., of Aarau, 
Switzerland, and this illustration is almost an exact representa- 
tion of the instruments used on the U. S. Lake and Mississippi 
River Surveys.^ The bubble is enclosed in a wooden case 
(metal case in the cut), and rests on top of the pivots or rings ; 

* The author had charge of about 600 miles of this work. 

f Called on that service geodesic levels. 

% This cut is from Fauth's Catalogue, Washington, D. C. 



GEODETIC SURVEYING, 



549 



it is carried in the hand when the instrument is transported. 
A mirror is provided which enables the observer to read the 
bubble without moving his eye from the eye-piece. There is a 
thumb-screw with a very fine thread under one wye which is 
used for the final levelling of the telescope when pointed on the 
rod. There are three levelling-screws, and a circular or box 
level for convenience in setting. The telescope bubble is very 




Fig. 149. 



delicate, one division on the scale corresponding to about three 
seconds of arc. The bubble-tube is chambered also, thus al- 
lowing the length of the bubble to be adjusted to different 
temperatures. The magnifying power is about 45 diameters. 
There are three horizontal wires provided, set at such a distance 
apart that the wire interval is about one hundredth of the dis- 
tance to the rod. The tripod legs are covered with white 



J50 SURVEYING. 



cloth to diminish the disturbing effects of the sun upon them. 
The level itself is always kept in the shade while at work. 

The levelling-rod is made in one piece, three metres long, of 
dry pine, about four inches wide on the face, and strengthened 
by a piece at the back, making a T-shaped cross-section. The 
rods are self-reading, that is, they are without targets, and are 
graduated to centimetres. An iron spur is provided at bottom 
which fits into a socket in an iron foot-plate. The end of the 
spur should be flat and the bottom of the socket turned out to 
a spherical form, convex upwards. A box-level is attached to 
the rod to enable the rodman to hold it vertically, and this in 
turn is adjusted by means of a plumb-line. Two handles are 
provided for holding the rod, and a wooden tripod to be used 
in adjusting the rod-bubble. The decimetres are marked on 
one side of the graduations and the centimetres on the other, 
all figures inverted since the telescope is inverting. 

403. The Instrumental Constants which must be accu- 
rately determined once for all, but re-examined each season, 
are — 

1. The angular value of one division on the bubble-tube. 

2. The inequality in the size of the pivot-rings. 

3. The angular value of the wire-interval, or the ratio of 
the intercepted portion on the rod to the distance of the rod 
from the instrument. 

4. The absolute lengths of the levelling-rods. 
These constants may be determined as follows: 

The value of one division of the bubble may be readily 
found by sighting the telescope on the rod, which is set at a 
known distance from the instrument, and running the bubble 
from end to end of its tube,Xaking rod-readings for each posi- 
tion of the bubble. The bubble-graduations are supposed to 
be numbered from the centre towards the ends. 



GEODETIC SURVEYING. 551 

Let E x = mean of all the eye-end readings of the bubble 
when it was run to the eye-end of its tube ; 
E t = same for bubble at object-end of tube ; 

1 = mean of all the object-end readings when bubble 

was at eye-end of tube ; 

2 = same for bubble at object-end of tube; 

R x = mean reading of rod for bubble at eye-end , 
R % = same for bubble at object-end ; 
D = distance from instrument to rod ; 
v = value of one division of the bubble (sine of the 
angle) at a unit's distance. 

- ^XX-o, « 

In seconds of arc we would have 

v (in seconds) = TWHT) F n\' • ( 2 ) 

D sin i" { x ~ l 



If a table is to be prepared for corrections to the rod-read- 
ings for various distances and deviations of the bubble from 
the centre of its tube, then the value as given by equation (i) 
is most convenient to use. The value of one division of a level 
bubble should be constant, but it is often affected by its rigid 
fastenings, which change their form from changes in tempera- 
ture. 

The inequality in the size of the rings is found by revers- 
ing the bubble on the rings, and also reversing the telescope 
in the wyes. The bubble is reversed only in order to eliminate 
its error of adjustment. The following will illustrate the 
method of making and reducing the observations: 



552 






SURVEYING. 


















Bubble-readings 






North. 




South. 


Tel. 


eye-end north. 


Lev. 


direct. 


4-3 




5-5 


«< 


<< «< 


(< 


reversed. 


4-7 
9.0 


(-1.7) 
— 0.42 


5.2 
IO.7 


Tel. 


eye-end south 


Lev. 


direct. 


6.2 




3-7 


« i 


<< <( 


K 


reversed. 


6.6 
12.8 


(+5.8) 
+ 1.45 


3.3 
7.0 


Tel. 


eye-end north 


Lev. 


direct. 


4.4 




5-5 


<t 


south 


<« 


reversed. 


il? 
9.2 


(-1.5) 
-0.38 


5-2 
10 7 



Mean reading north = — 0.40 

" " south = + 1 .45 

North minus south = — 1.85 



That is to say, the bubble moves 1.85 divisions towards the 
object-end when the telescope is reversed in the wyes. This is 
evidently twice the inequality of the pivot-rings ; and since the 
axis of a cone is inclined to one of its elements by one half 
the angle at the apex, so the line of sight is inclined to the 
tops of the rings by one fourth of 1.85 divisions, or 0.46 divi- 
sions of the bubble. It is also evident that the eye-end ring 
is the smaller, and that therefore when the top surfaces of the 
rings are horizontal the line of sight inclines downward from 
the instrument. The correction is therefore positive. This is 
called the pivot-correction, and changes only with an unequal 
wear in the pivot-rings. 

The angular value of the wire-interval is found by measur- 
ing a base on level ground of about 300 feet from an initial 
point c-\-f* in front of the objective. Focus the telescope 
on a very distant object, and measure the distance from the 
inside of the objective to the cross-wires, this being the value 



* See art. 205 for the significance of these terms, as well as for the theory 
of the problem. 



GEODETIC SURVEYING. 553 



of /for that instrument. Measure the space intercepted on 
the rod between the extreme cross-wires. 



If d = length of base, counting from the initial point ; 
s = length of the intercepted portion of the rod ; 

/ 

r — - = constant ratio of distance to intercept ; 

d 

then r = — ; 
s 



and for any other intercept s f on the rod we have 

d' = rs>+f+c (3) 

When r,f, and c are found, a table can be prepared giving 
distances in terms of the wire-intervals. 

The errors in the absolute lengths of the rods affect only 
the final differences of elevation between bench-marks. This 
correction is usually inappreciable for moderate heights. 

404. The Daily Adjustments. — The adjustments which 
are examined at the beginning and close of each day's work 
are as follows : 

1. The collimation, that is, the amount by which the line 
of sight, as determined by the mean reading of the three wires, 
deviates from the line joining the centres of the rings. 

2. The bubble-adjustment — that is, the inclination of the 
axis of the bubble to the top surface of the rings. 

3. The rod-level. This is examined only at the beginning 
of each day's work, and made sufficiently perfect. 

The first two adjustments are very important, since it is by 
means of these (in conjunction with the pivot-correction, 
determined once for the season) that the relation of the bubble 



554 SURVEYING. 



to the line of sight is found. It is not customary in this work 
to try to reduce these errors to zero, but to make them reason- 
ably small, and then determine their values and correct for 
them. It is evident that if the back and fore sights be kept 
exactly equal between bench-marks, then the errors in the 
instrumental adjustments are fully eliminated ; and in any case 
these errors can only affect the excess in length of the sum of 
the one over that of the other. It is to this excess in length 
of back-sights over fore-sights, or vice versa, that the instru- 
mental constants are applied ; but in order to apply them their 
values must be accurately determined. The curvature of a 
level surface would also enter into this excess, but it is usually 
so small a residual distance, that the correction for curvature 
is quite insignificant. There are, however, three instrumental 
corrections to be applied for the amount of the excess, namely, 
the corrections for collimation, inclination of bubble, and in- 
equality of pivots, designated respectively by c, i, and/. Since 
three horizontal wires are read on the rod, the wire-intervals 
can be used in place of the distances, for they are linear func- 
tions practically, and so a record is kept of the continued sum 
of the lengths of the back and fore sights, and from these the 
final difference is found. 

The collimation-correction is taken out for a distance of 
one unit (the metre has been universally used in this kind of 
levelling), and then the correction for any given case found by 
multiplying by the residual distance. 

Let R x = rod-reading for telescope normal ; 

R^= " " " " inverted; 

d = distance of rod from instrument. 



Then <^=~A W 



2d 



GEODETIC SURVEYING. 555 

The correction for the inclination of the bubble to the tops 
of the rings is found by reversing the bubble on the telescope 
and reading it in both positions. In such observations the 
initial and final readings are taken with the bubble in the same 
position, thus giving an odd number of observations. Usually 
two direct and one reversed reading are taken. The correction 
is found in terms of divisions on the bubble, the correction in 
elevation being taken from the table prepared for that purpose. 

" et E x == mean of the eye-end* readings for level direct ; 
£ 2 — " " " " " " reversed ; 

0,= " " . object " " " direct ; 

<? 3 = " " " " " " reversed ; 

,.L(i=fl.4^a) M 

The pivot correction has already been found, and is sup- 
posed to remain constant for the season. 

If E be the excess of the sum of the back-sights over that 
of the fore-sights, then the final correction for this excess is 

C^E\c + •(*■■+/)], (3) 

where v is taken from eq. (i), p. 551. Evidently, if the fore- 
sights are in excess, the correction is of the opposite sign. 

405. Field Methods. — The great accuracy attained in pre- 
cise levelling is due quite as much to the methods used and 
precautions taken in making the observations as to the instru- 
mental means employed. Aside from errors of observation 
and instrumental errors, we have two other general classes of 

* By eye-end is always meant the end towards the eye-end of the telescope, 
whether in a direct or a reversed position. 



556 SURVEYING. 



errors, which can be avoided only by proper care being used 
in doing the work. These two classes are errors from unstable 
supports and atmospheric errors. 

Any settling of the rod between the fore and back readings 
upon it will result in the final elevation being too high, while 
any settling of the instrument between the back and fore 
readings from it will also result in too high a final elevation. 
Such errors are therefore cumulative, and the only way in 
which they can be eliminated is to duplicate the work over 
the same ground in the opposite direction. As a general pre- 
caution, the duplicate line should always be run in the opposite 
direction. This will result in larger discrepancies than if both 
are run in the same direction, but the mean is nearer the truth. 

Atmospheric errors may come from wind, heated air-cur- 
rents causing the object sighted to tremble or " dance," or 
from variable refraction. For moderate winds the instrument 
may be shielded by a screen or tent, but if its velocity is more 
than eight or ten miles an hour, work must be abandoned. 
To avoid the evil effects of an unsteady atmosphere the length 
of the sights is shortened ; but when a reading cannot be well 
taken at a distance of about 150 feet, or 50 metres, it would 
be better to stop, since the errors arising from the number of 
stations occupied would make the work poor. At about 8 
o'clock A.M. and 4 P.M. very large changes in the refraction 
have been observed on lines over ground which is passing from 
sun to shade, or vice versa, when the image was apparently 
very steady. In clear weather not more than three or four 
hours a day can be utilized for the best work, and sometimes, 
with hot days and cool nights, it is impossible to get an hour 
when good work can be done. 

In making the observations the bubble is brought exactly 
to the centre of its tube, the observer being able to do this 
by means of the thumb-screw under one wye, and the mirror 
which reflects the image of the bubble to the observer at the 



GEODETIC SURVEYING. 557 



eye-piece. If there is no mirror to the bubble, then it is 
brought approximately to the centre, and the recorder reads 
it while the observer is reading the three horizontal wires. In 
any case the bubble-reading is recorded in the note-book, and 
if it was not in the middle a correction is made for the eccen- 
tric position by means of a table prepared for the purpose. 
The mean of the three wire-readings is taken as the reading 
of that rod, the observer estimating the tenths of the centi- 
metre spaces, thus reading each wire to the nearest millimetre. 
The wires should be about equally spaced so that the mean of 
the three wires coincides very nearly with the middle wire. 
The differences between the middle and extreme wire-readings 
are also taken out to give the distance, as well as to check the 
readings themselves by noting the relation of the two intervals. 
If they are not about equal, then one or more of the three 
readings is erroneous. This is a most important check, and 
constitutes an essential feature of the method. 

It has been found economical to have two rodmen to each 
instrument, so that no time shall be lost between the back and 
fore sight readings from an instrument-station. Since but a 
small portion of the day can generally be utilized, it is desira- 
ble to make very rapid progress when the weather is favora- 
ble. When two rodmen are used, and the air is so steady that 
100-metre sights can be taken,* it is not difficult for an expe- 
rienced observer to move at the rate of a mile an hour. 

On the U. S. Coast and Geodetic Survey a much more 
laborious method of observing than the one above outlined 
has been followed. There a special kind of target-rod has 
been employed, the target being set approximately and 
^clamped. The thumb-screw under the wye is used as a mi- 
crometer-screw, and two readings are taken on it one when 



* This is about the limiting length of sight for first-class work, even under 
the most favorable conditions. 



553 SURVEYING. 



the bubble is in the middle and the other when the centre 
wire bisects the target,* the bubble now not being in the 
middle, since the target's position was only approximate. The 
bubble is then reversed, and two more readings of the screw 
taken. The telescope is now revolved in the wyes, and read- 
ings taken again with bubble direct and reversed. Thus there 
are four independent readings taken on the rod, each necessi- 
tating two micrometer-readings. The reduction is also very 
complicated, each sight being corrected for curvature and re- 
fraction as well as for instrumental constants. The duplicate 
line is carried along with the first one by having two sets of 
turning-points for each instrument-station. The instrument, 
however, is set but once, so that the lines are not wholly inde- 
pendent. The alternate sections are run in opposite directions, 
thus partly obviating the objection to running both lines in 
the same direction. The method first described was used on 
the U. S. Lake and Mississippi River surveys, and is also the 
method used on most of the European surveys of this char- 
acter. 

The instrument is always shaded from the sun, both while 
standing and while being carried between stations. It is abso- 
lutely necessary to do this in order to keep the adjustments 
approximately constant, and the bubble from continually 
moving. 

406. Limits of Error. — On the U. S. Coast and Geodetic 
Survey the limit of discrepancy between duplicate lines is 
5 mm V2K where K is the distance in kilometres. On the U. S. 
Lake Survey the limit was io mm VK, and on the Mississippi 
River Survey it was 5 mm VK. These limits are respectively 
0.029. 0.041, and 0.021 feet into the square root of the distance ' 
in miles. If any discrepancies occurred greater than these the 
stretch had to be run again. 

The " probable error" of the mean of several observations 
on the same quantity is a function of the discrepancies of the 



GEODETIC SURVEYING. 559 

several results from the mean. If v v v 2 , v z , etc., be the several 
residuals obtained by subtracting the several results from the 
mean, and if 2\yv\ be the sum of the squares of these residu- 
als, and if m be the number of observations, then the probable 

error of the mean is R = ± .6745 A / — _i!^L . 

y rniin — 1) 

This is the function which is universally adopted for meas- 
uring the relative accuracy of different sets of observations. 
If there be but two observations this formula reduces to 

R=±W, 

where Fis the discrepancy between two results. 

The European International Geodetic Association have 
fixed on the following limits of probable error per kilometre 
in the mean or adopted result: + 3 mm per km. is tolerable; 
± 5 mm per km. is too large ; ± 2 mm per km. is fair ; and ± i mra 
per km. is a very high degree of precision. On the U. S. 
coast and geodetic line from Sandy Hook to St. Louis, a dis- 
tance of 1009 miles, the probable error per kilometre was 
± l.2 mm .* For the 670 miles of this work on the Mississippi 
River Survey, of which the author had charge, the probable 
error of the mean for the entire distance was 23.5 mm (less than 
one inch), and the probable error per kilometre was ± 0.7 mm .t 
Of course very little can be predicated on these results as to the 
actual errors of the work, since the number of observations on 
each value was usually but two ; but they may fairly be used 
for the purpose of comparing the relative accuracy of different 
lines where this function has been computed from similar 
data. 

407. Adjustment of Polygonal Systems in Levelling. — If 

* Report U. S. Coast and Geodetic Survey, 1882, p. 522. 

f Reports of the Miss. Riv. Commission for the years 1882, 1883, and 1884. 



560 SURVEYING. 



a line of levels closes upon itself the s'ummation of all the differ- 
ences of elevation between successive benches should be zero. 
If it is not, the residual error must be distributed among the 
several sides, or stretches, composing the polygon, according 
to some law, so that the final corrections which are applied to 
the several sides shall be independent of all personal considera- 
tions. These corrections should also be the most probable 
corrections. There are two general criterions on which to 
found a theory of probabilities. One may be called a priori, 
and the other a posteriori. By the former we would say that 
the errors made are some function of the distance run, as that 
they are directly proportional to this distance, or to the square 
root of this distance, etc.; while by the latter, or a posteriori 
method, we would say the errors made on the several lines are 
a function of the discrepancies found between the duplicate 
measurements on those lines, or to the computed " probable 
error per kilometre," as found from these discrepancies. Both 
methods are largely used in the adjustment of observations. 
These laws of distribution are equivalent to establishing a 
method of weighting the several sides of the system, a larger 
weight implying that a larger share of the total error is to be 
given to that side. When any system of weights is fixed upon, 
then the corrections may be computed by the methods of least 
squares so as to comply with the condition that the corrections 
shall be the most probable ones for that system of weighting. 
The most probable set of corrections is that set the sum of 
whose squares is a minimum. If the system includes more 
than a few polygons, this method of reduction is exceedingly 
laborious, while the increased accuracy is very small over that 
from a much simpler method. 

Fig. 150 represents the Bavarian network of geodetic levels, 
there being four polygons. Every side has been levelled, and 
the difference of elevation of its extremities found. These ele- 
vations must now be adjusted so that the differences of eleva- 



GEODETIC SURVEYING. 



5 6l 




Fig. 150. 



tion on each polygon shall sum up zero. When these sums 
are taken the following residuals are found : I., -|- 20.2 mm ; II., 
+ 39-3 mm l HI., — 25.2 mm ; and 
IV., + io8.0 mm . It was sup- 
posed that an error of one deci- 
metre had been made in the 
fourth polygon, but in the ab- 
sence of any knowledge in the 
case this error must be distrib- 
uted with the rest. 

The method which the au- 
thor would recommend is a 
modification of Bauernfeind's, 
iri that the errors are to be made 
proportional to the square roots of the lengths of the sides in- 
stead of the lengths of the sides directly. Since the errors in 
levelling are compensating in their nature they would be ex- 
pected to increase with the square root of the length of the 
line, and it is the author's experience that the error is much 
nearer proportional to the square root of the distance than to 
the distance itself. 

Instead of treating the four polygons as one system and 
solving by least squares, the polygon having the largest error 
of closure is first adjusted by distributing the error among its 
sides in proportion to the square roots of the lengths of those 
sides. Then the polygon having the next largest error is ad- 
justed, using the new value for the adjusted side, if it is con- 
tiguous to the former one, and distributing the remaining 
error among the remaining sides of the figure, leaving the 
previously adjusted side undisturbed. The adjustment pro- 
ceeds in this manner until all the polygons are adjusted. The 
Bavarian system is worked out on this plan in the following 
tabulated form : 



562 



SURVEYING. 



ADJUSTMENT OF THE BAVARIAN SYSTEM OF LEVEL 

POLYGONS. 



No. 
Side. 


Length. 


Sq. Root 

of 

Length 

= A. 


No. 
Polygon. 


2\. 


Difference 

of 
Elevation. 

m. 


Error 

of 

Closure 

mm. 


Cor- 
rected 
Error 
of 
Closure 


Cor- 
rection. 


Corrected 
Difference 

of 
Elevation. 




km. 












1 


125.8 


11 


2 


I. 


24.6 


+ 35-8723 


-f- 20.2 


+ 3i-3 


- 14-3 


+ 35-858o 


2 


179.0 


13 


4 


I. 




— 217.5062 






— 17.0 


— 217.5232 


3 


T 47-3 


12 


1 


II. 




± 181. 6541 


+ 39-3 


+ 39-3 


— 11. 1 


± 181.6652 


4 


60.6 


7 


8 


II. 


43- 1 


-f 3 2 -°958 






— 7.1 


+ 32.0887 


5 


174.0 


13 


2 


II. 




+ i79-598i 






— 12.0 


+ i79-586i 


6 


IOI.I 


10 





II. 


20.9 


T 30 . 0005 


— 25.2 


+ 19-9 


- 9.1 


T 30.0096 


7 


134-9 


11 


6 


III. 




— 38.6644 






— 11. 


— 38.6754 


8 


80.1 


9 





IV. 




T 48.8053 






— 36.0 


± 48.7 6 93 


9 


87.0 


9 


3 


III. 




4- 57-4440 






- 8.9 


+ 57-435 1 


10 


96.8 


9 


8 


IV. 


27.0 


— 100.1619 


+ 108.0 


+ 108.0 


- 39-2 


— 100.2011 


11 


67.9 


8 


2 


IV. 




-4- 51.4646 






- 32.8 


+ 5i-43 l8 



Beginning with polygon IV., we find its error of closure to 
be -f- io8.o mm , this being distributed among the three sides so 
that -f^Q goes to side 8, -^ to side 10, and -£fc to side 11. 
The corrected values for these sides are now found. Next 
take the polygon having the next largest error of closure, 
which is number II., and distribute its error in like manner. 
This leaves polygons I. and III. to be adjusted, one side of 
the former and two of the latter being already adjusted. The 
corrected errors of closure for these polygons are 3 r.3 mm and 
I9.9 mm respectively, the former to be distributed between the 
sides I and 2 and the latter between the sides 7 and 9. The 
resulting corrected values cause all the polygons to sum up 
zero. 

The sum of the squares of the corrections here found is 
50.02 square centimetres, whereas if the differences of eleva- 
tion had been weighted in proportion to the lengths of the 
sides and the system adjusted rigidly by least squares the sum 
of the squares of the corrections would have been 49.65 square 
centimetres, showing that the method here used is practically 



GEODETIC SURVEYING. 563 

as good as the rigid method which is commonly used. It has 
been found in practice to give, in general, about the same 
sized corrections as the rigid system. 

408. Determination of the Elevation of Mean Tide. — 
To determine accurately the elevation of mean tide at any 
point on the coast requires continuous observations by means 
of an automatic self-registering gauge for a period of several 
years. The methods of making these observations with cuts 
of the instruments employed are given in Appendix No. 8 of 
the U. S. Coast Survey Report for 1876. A float, inclosed in 
a perforated box, rises and falls with the tide, and this motion, 
properly reduced in scale by appropriate mechanism, is re- 
corded by a pencil on a continuous roll of paper which is moved 
over a drum at a uniform rate by means of clockwork. An 
outer staff-gauge is read one or more times a day by the at- 
tendant, who records the height of the water and the time of 
the observation on the continuous roll. This outer staff is 
connected with fixed bench-marks in the locality by very 
careful levelling, and this connection is repeated at intervals to 
test the stability of the gauge. 

To find from this automatic record the height of mean tide, 
ordinates are measured from the datum-line of the sheet to 
the tide-curve for each hour of the day throughout the entire 
period. This period should be a certain number of entire 
lunar months. The mean of all the hourly readings for the 
period maybe taken as mean tide. It maybe found advisable 
to reject all readings in stormy weather, in which case the 
entire lunation should be rejected. 



CHAPTER XV. 

PROJECTION OF MAPS, MAP-LETTERING, AND TOPO- 
GRAPHICAL SYMBOLS. 

I. PROJECTION OF MAPS. 

409. THE particular method that should be employed in 
representing portions of the earth's surface on a plane sheet' 
or map depends, first, on the extent of the region to be repre- 
sented ; second, on the use to be made of the map or chart ; 
and third, on the degree of accuracy desired. 

Thus, a given kind of projection may suffice for a small 
region, but the approximation may become too inaccurate 
when extended over a large area. It is quite impossible to 
represent a spherical surface on a plane without sacrificing 
something in the accuracy of the relative distances, courses, 
or areas ; and the use to which the chart is to be put must de- 
termine which of these conditions should be fulfilled at the 
expense of the others. A great many methods have been 
proposed and used for accomplishing various ends, some of 
which will be described. 

410. Rectangular Projection. — In this method the merid- 
ians are all drawn as straight parallel lines ; and the parallels 
are also straight, and at right angles with the meridians. A 
central meridian is drawn, and divided into minutes of latitude 
according to the value of these at that latitude as given in 
Table XI. Through these points of division draw the paral- 
lels of latitude as right lines perpendicular to the central 
meridian. On the central parallel lay off the minutes of 



PROJECTION OF MAPS. 565 

longitude, according to their value for the given latitude, by 
Table XL; and through these points of division draw the other 
meridians parallel with the first. y 

The largest error here is in assuming the meridians to be 
parallel. For the latitude of 40 , two meridians a mile apart 
will converge at the rate of about a foot per mile. A knowl- 
edge of this fact will enable the draughtsman to decide when 
this method is sufficiently accurate for his purpose. Thus, for 
an area of ten miles square, the distortion at the extreme cor- 
ners in longitude, with reference to the centre of the map as 
an origin of coordinates, will be about twenty-five feet. At 
the equator this method is strictly correct. 

In this kind of projection, whether plotted from polar or 
rectangular coordinates, or from latitudes and longitudes, all 
straight lines of the survey, whether determined by triangula- 
tion or run out by a transit on the ground, will be straight on 
the map ; that is, the fore and back azimuth of a line is the 
same, or, in other words, a straight line on the drawing gives 
a constant angle with all the meridians. 

This is the method to use on field-sheets, where the survey 
has all been referred to a single meridian. 

411. Trapezoidal Projection. — Here the meridians are 
made to converge properly, but both they and the parallels 
are straight lines. A central meridian is first drawn, and grad- 
uated to degrees or minutes ; and through these points paral- 
lels are drawn, as before. Two of these parallels are selected ; 
one about one fourth the height of the map from the bottom, 
and the other the same distance from the top. These paral- 
lels are then subdivided, according to their respective lati- 
tudes, from Table XL ; and through the corresponding points 
of division the remaining meridians are drawn as straight lines. 
The map is thus divided into a series of trapezoids. The 
parallels are perpendicular to but one of the meridians. The 
principal distortion comes from the parallels being drawn as 



566 SUlt VE YING. 



straight lines, and amounts to about thirty-two feet in ten 
miles in latitude 40 , and is nearly proportional to the square 
of the distance east or west from the central meridian. 

The work should be plotted from computed latitudes and 
longitudes. The method is adapted to a scheme which has a 
system of triangulation for its basis, the geodetic position of 
the stations having been determined. These conditions would 
be fulfilled in a State topographical or geological survey for 
the separate sheets, each sheet covering an area of not more 
than twenty-five miles square. 

412. The Simple Conic Projection. — In this projection, 
points on a spherical surface are first projected upon the sur- 
face of a tangent cone, and then this conical surface is devel- 
oped into the plane of the map. The apex of the cone is 
taken in the extended axis of the earth, at such an altitude 
that the cone becomes tangent to the earth's surface at the 
middle parallel of the map. When this conical surface is de- 
veloped into a plane, the meridians are straight lines converg- 
ing to the apex of the cone, and the parallels are arcs of con- 
centric circles about the apex as the common centre. 

The sheet is laid out as follows: Draw a central meridian, 
and graduate it to degrees or minutes, according to their true 
values a*s given in Table XI. Through these points of divi- 
sion draw parallel circular arcs, using the apex of the cone as 
the common centre. For values of the length of the side of 
the tangent cone, which is the length of the central parallel 
above, see Table XI. The rectangular coordinates of points 
in these curves are also given in the same table. 

On the middle parallel of the map the degrees or minutes 
of longitude are laid off, and through these are drawn the re- 
maining meridians as straight lines radiating from the apex 
of the tangent cone. 

It will be seen that the latitudes are correctly laid off, and 
the degrees of longitude will be sufficiently accurate for a map 



PROJECTION OF MAPS. 567 

covering an area of several hundred miles square. The merid- 
ians and parallels are at right angles. 

In this projection the degrees of longitude on all parallels, 
except the middle one, are too great ; and therefore the area 
represented on the map is greater than the corresponding area 
on the sphere. 

The chart should be plotted from computed latitudes and 
longitudes. 

413. De Tlsle's Conic Projection. — This is very similar 
to the above, except that two parallels, one at one fourth, and 
one at three fourths the height of the map, are properly grad- 
uated, and the meridians drawn as straight lines through these 
points of division. The parallels are drawn as concentric cir- 
cles, as in the simple conic projection. This is therefore but a 
combination of the second and third methods, and is more 
accurate than either of them. The cone here is no longer tan- 
gent, but intersects the sphere in the two graduated parallels. 
In this case the region between the parallels of intersection is 
shown too small, and that outside these lines is shown too 
large ; so that the area of the whole map will correspond very 
closely to the corresponding area on the sphere. When these 
parallels are so selected that these areas will be to each other 
exactly as the scale of the drawing, then it is called " Mur- 
doch's projection." 

414. Bonne's Projection. — This differs from the simple 
conic in this — that all the parallels are properly graduated, 
and the meridians drawn to connect the corresponding points 
of division in the parallels. These latter are, however, still 
concentric circles. The meridians are at right angles to the 
parallels only in the middle portion of the map. The same 
scale applies to all parts of the chart. There is a slight dis- 
tortion at the extreme corners, from the parallels being arcs 
of concentric circles. The proportionate equality of areas is 






568 SURVEYING. 



preserved. A rhumb-line appears as a curve ; but when once 
drawn, its length may be properly scaled. 

It will be noted that the last three methods involve the 
use of but one tangent or intersecting cone. 

415. The Polyconic Projection. — For very large areas it 
is preferable to have each parallel the development of the 
base of a cone tangent in the plane of the given parallel. 
This projection differs from Bonne's only in the fact that the 
parallels are no longer concentric arcs, but each is drawn with 
a radius equal to the side of the cone which is tangent at 
that latitude. These, of course, decrease towards the pole ; 
and therefore the parallels diverge from each other towards 
the edge of the chart. The result of this is, that a degree 
of latitude at the side of the map is not equal to a degree 
on the central meridian ; or, in other words, the same scale 
cannot be applied to all parts of the map. These defects ap- 
pear, however, only on maps representing very large areas. 
The whole of North America could be represented by this 
method without any material distortion. 

This method of projection is exclusively used on the Unit- 
ed States Coast and Geodetic Survey, and for all other maps 
and charts of large areas in this country. Extensive tables are 
published by the War and Navy Departments, and also by 
the Coast Survey, to facilitate the projection of maps by the 
polyconic system. Table XL gives in a condensed form the 
rectangular coordinates of the points of intersection of the 
parallels and meridians referred to the intersection of the sev- 
eral parallels with the central meridian as the several origins. 

416. Formulae used in the Projection of Maps.* — The 
fundamental relations on which the method of polyconic pro- 
jection rests are given in the following formulae: 

* See Appendix D for the derivation of equations (1) and (2). 



PROJECTION OF MAPS. 569 

Normal, being the radius of curvature 
of a section perpendicular to the 

meridian at a given point N = -, a e . „ ~. (1) 

(1 — e sin Z)* v J 

where R e is the equational radius, 
e is the eccentricity, 
and L is the latitude. 



(l _ e *\ 

Radius of the meridian R m = N - — ni— • • (2) 

i\. e 

Radius of the parallel R p = N cos L. . . (3) 



7t 

Degree of the meridian D m = ~—-R m ... (4) 

I bO v J 

= $6ooR m sin 1". 



7t 

Degree of the parallel Dp = —^-Rp ... (5) 

= $6ooRp sin 1". 

Radius of the developed parallel, or 

side of tangent cone r = N cot L. . . . (6) 

If # be any arc of a parallel, in degrees, or any difference 
of longitude from the central meridian of the drawing, and 
if 6 be the corresponding angle, in degrees, at the vertex of 
the tangent cone, subtended by the developed parallel, then 
since the angular value of arcs of given lengths are inversely 
as their radii, we have 

e Rp 

- = —^ = sin L, 
n r 

or 6 = 71 sin L (8) 



570 SURVEYING. 



Since the developed parallels are circular arcs, the rectangu- 
lar coordinates of any point an angular distance of 6 from 
the central meridian is, 



Meridian distance, d m = x = r sin 6. 

Divergence of parallels, dp = y = r vers 6. y . , (9) 

= x tan \Q> 



For arcs of small extent, the parallel may be considered 
coincident with its chord ; but the angle between the axis of x 
and the chord is \d. If, then, the length of the arc, which is 
nDp, be represented by the chord, we may write 



d m = meridian distance = x = iiDp cos \6, r 

and dp = divergence of parallels = y = nDp sin -J0. ' 



If, now, d m * = meridian distance for 1 degree of longitude, 
and d mn = meridian distance for n degrees of longitude, 



d mn nDp cos \B n 

we have -j — = -ft rs~* 

d mx Dp cos \V X 



But = n sin L, so that O x = i° X sin L = 38' for latitude 40°. 
Therefore 



cos $0 X = cos 19' = I, nearly; 



so that -?2 = n cos \{ri sin L), nearly (n) 



PROJECTION OF MAPS. 571 

For L = 30 , we have sin L = \. Therefore, for latitude 30 , 
-~ = n cos \n = n cos (0.25/2), nearly. 

If we have obtained the meridian distance, d my for 1 degree 
of longitude, and wish to obtain it for n degrees in latitude 
30 , we have but to multiply the distance for 1 degree by n 
cos (0.25/2). 

417. In Table XI. the meridian distances are given, at vari- 
ous latitudes, for a difference of longitude of one degree. To 
find the meridian distance for any number of degrees or parts 
of degrees, multiply the distance for one degree by the factor 
there given for the given latitude. The factor given in the 
table for latitude 30 is n cos (0.288/2), in place of n cos (0.25/2), 
as obtained above. The difference is a correction which has 
been introduced to compensate the error made in assuming 
that the chord was equal in length to its arc. The corrected 
factors enable the table to be used without material error up 
to 25 degrees longitude either side of the central meridian. 

To obtain the divergence of the parallels for differences of 
longitude more or less than one degree, multiply the diver- 
gence for one degree by the square of the number of degrees. 
It is evident that this rule is based on the assumption that the 
arc of the developed parallel is a parabola, and so it may be 
considered for a distance of 25 degrees either side of the cen- 
tral meridian between the latitudes 30 and 50 without mate- 
rial error. 

If the whole of the United States were projected by this 
table, using the factors given, to a scale of I to 1,500,000, thus 
giving a map some 8 by 10 feet, the maximum deviation of 
the meridians and parallels from their true positions (which 
would be at the upper corners) would be but about 0.02 inch. 



572 



SUR VE YING. 




Fig. 151. 



Thus, for a map of this size, covering 20 degrees of lati- 
tude and 50 degrees of longitude, the geodetic lines would 

have their true position within the 
width of a fine pencil line, by the 
use of Table XL Fig. 151 will illus- 
trate the use of the table in project- 
ing a map by the polyconic method. 
The map covers 30 degrees in lati- 
tude (30 to 6o°) and 60 degrees in 
longitude. The straight line 3 6 is 
first drawn through the centre of the map, and graduated ac- 
cording to the lengths of one degree of latitude, as given in 
the second column of Table XL Through these points of di- 
vision the lines m\m 3 , are drawn in pencil at right angles to 
the central meridian. On these lines the points m v m v etc., 
are laid off by the aid of the first part of Table XL This ta- 
ble gives the meridian distances when n is less than one degree, 
as well as when n is greater. From the points m v m„ etc., 
the divergence of the parallels is laid off above the lines Om, 
by the aid of the second portion of Table XL, thus obtaining 
the positions of the points p x , p„ etc. The points/ mark the 
intersection of the meridians and parallels ; and these may 
be drawn as straight lines between these points, provided a 
sufficient number of such points have been located. The map 
is then to be plotted upon the chart irom computed latitudes 
and longitudes. 

418. Summary. — We have seen that there are, in general, 
two ways of plotting a map or chart, and two corresponding 
uses to which it may be put: 

First. We may plot by a system of plane coordinates, 
either polar (azimuth and distance) or rectangular (latitudes 
and departures). This gives a map from which distance, 
azimuth (referred to the meridian of the map), and areas are 
correctly determined. 



> 



PROJECTION OF MAPS. 573 

Second. We may plot the map by computed latitudes and 
longitudes, and determine from it the relative position of points 
in terms of their latitude and longitude. 

The first system is adapted to small field sheets and detail 
charts for which the notes were taken by referring all points 
to a single point and meridian. For this purpose the system 
of rectangular projection should be selected, as long as the 
area of the chart is not more than about one hundred square 
miles. If it be larger than this, the trapezoidal system should 
be used. In case this is done, the work is still plotted as 
before, provided it has all been referred to a given meridian in 
the field work, and then converging meridians are drawn as 
described above. From such a chart, not only the azimuth 
(referred to the central meridian) and distance may be deter- 
mined, but the correct longitude and nearly correct latitude 
are given. 

In the case of topographical charts, based on a system of 
triangulation, each sheet is referred to a meridian passing 
through a triangulation-station on that sheet, or near to it, 
and the rectangular system used. 

In the case of a survey of a long and narrow belt, as 
for a river, railroad, or canal, if the survey was based on a 
system of triangulation, the convergence of meridians has been 
looked after in the computation of the geodetic positions of 
these stations, and each sheet is plotted by the rectangular 
system, being referred to the meridian through the adjacent 
triangulation-station. When many of these are combined into 
a single map on a small scale, then they must be plotted on 
the condensed map by latitudes and longitudes, these being 
taken from the small rectangular projections, and plotted on 
the reduced chart in polyconic projection ; the meridians and 
parallels having been laid out as shown above. 

In case the belt extends mostly east and west, and is not 
based on a triangulation scheme, then observations for azimuth 



5^4 SURVEYING. 



should be made as often as every fifty miles. It will not do 
to run on a given azimuth for this distance, however ; for there 
has been a change in the direction of the parallel (or meridian) 
in this distance, in latitude 40 , of about 40 minutes. Accord- 
ing to the accuracy with which the work is done, therefore, 
when running wholly by back azimuths, the setting of the in- 
strument must be changed. Thus, if in going 1 degree (53 
miles), east or west, in latitude 40°, the meridian has shifted 
4c/, then in going 13 miles east or west the meridian has 
changed io'; and this is surely a sufficiently large correction 
to make it worth while to apply it. 

When running west, this correction is applied in the direc- 
tion of the hands of a watch, and when running east, in the 
opposite direction; that is, having run. west 13 miles by back 
azimuth, then the pointing which appears north is really io' 
west of north, and the telescope must be shifted io' around to 
the right. 

If the azimuth be corrected in this way, a survey can be 
carried by back azimuth an indefinite distance, and still have 
the entire survey referred to the true meridian. 

419. The Angle of Convergence of Meridians is the 
angle 6 in the equations given in the above formula. Then 

6 = n sin L, 

where n is the angular change in degrees of longitude, and L 
is the latitude of the place. 

For L = 30 , sin L = i; or, in latitude 30 a change of 
longitude of one degree changes the direction of the meridian 
by 30 minutes. 

For L = 40 , sin L == O.643 ; or, a change of longitude of 
one degree changes the direction of the meridian by 0.643 of 
60 minutes, or 38.6 minutes, being approximately 40 minutes. 

For L = 50 , sin Z= 0.766; or, in going east or west one 



MAP-LETTERING AND TOPOGRAPHICAL SYMBOLS. 575 

degree, the meridian changes 0.766 X 60 minutes = 46 min- 
utes, or approximately 50 minutes. 

Therefore we may have the approximate rule, that a change 
of longitude of one degree changes the azimuth by as many 
minutes as equals the degrees of latitude of the place. This 
rule gives results very near the truth between plus and minus 
40 latitude, that is, over an equatorial belt 80 degrees in 
width. 

II. MAP-LETTERING AND TOPOGRAPHICAL SYMBOLS. 

420. Map-Lettering. — The best-drawn map may have its 
appearance ruined by the poor skill or bad taste displayed in 
the lettering. The letters should be simple, neat, and dignified 
in appearance, and have their size properly proportioned to the 
subject. The map is lettered before the topographical symbols 
are drawn. When a map is drawn for popular display, some 
ornamentation may be allowed in the title ; but even then, 
the lettering on the map itself should be plain and simple. 
When the map is for official or professional use, even the title 
should be made plain. 

On Plate IV. are given several sets of alphabets which are 
well adapted to map work. Of course the size should vary 
according to the scale of the map and the subject, as shown on 
Plate V. It is a good rule to make all words connected with 
water in italics. The small letters in stump writing will be 
found very useful, and these should be practised thoroughly. 
The italic capitals go with these small letters also. 

In place of the system of letters above described, and 
which has heretofore been almost exclusively used for map- 
ping purposes, a new system, called " round writing," may be 
used. A text-book on this method, by F. Soennecken, is pub- 
lished by Messrs. Kueffel & Esser, New York. This work is 
done with blunt pens, all lines being made with a single stroke. 



576 SURVEYING. 



It looks well when well done, and requires but a small fraction 
of the time required to make the ordinary letters. For work- 
ing drawings and field maps it is especially adapted. 

421. Topographical Symbols. — In topographical repre- 
sentation, where elevations have been taken sufficiently num- 
erous and accurate, the outline of the ground should be rep- 
resented by contours rather than by hachures, or hill shading, 
which simply gives an approximate notion of the slope of the 
ground, but no indication of its actual elevation. Where the 
ground has so steep a slope that the contour lines would fall 
one upon another, it is well here to put in shading-lines, as 
shown on Plate III. The water surfaces and streams may be 
water-lined in blue, or left white. The contour lines over -al- 
luvial ground should be in brown (crimson and burnt sienna), 
while those over rocky and barren ground should be in black. 
This is a very simple and effective method of showing the 
character of the soil. 

The practices of the government surveys should be fol- 
lowed in the matter of conventional surface representation, 
such as meadow, swamp, woodland, prairie, cane-brake, etc., 
with all their varieties. Some of these are given in the United 
States Coast Survey Report for 1879 and 1883, while Plate III. 
shows most of those used on the Mississippi River Survey. 
Those shown in Plate II. are adapted to higher latitudes, and 
are those used in the field-practice surveys at Washington 
University. This plate is an exact copy of one of the annual 
maps made from actual surveys by the Sophomore class. On 
these the contours are all in black, for the purpose of photo- 
lithographing. 



PLATE 1. 




i' 




ISOCONIC CHART FOR 1885. 






ft 



Conventional Signs For 
TOPOGRAPHICAL MAPS, 

SCALE 1 : loiKKi 

Designed ["or Photo -lithographing. 



PLATE III. 




Di.ovn and engraved l>\ 
EDWARD MOL1T0R. T.Ej Si Louis 







TR1ANGUJLATICW SYSTEM. 

Sn(i]e:-15(10 Dtt = one nn-Ji 



Ft 1 ■ , |^, 





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I ■ ! 


i 


»"'"" IIJI ■ 


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TOPOOnAPinCAL PR ACTICE SURVEY 

SWEET SPP LINGS MO. 

i., thl i 
SOPHOMORE CLASS 

in (He 

POLYTECHNIiC SCHOOL 

of 

WASHINGTON UNIVERSITY 
18816 

Drawn t iy 
i H i Wifl « 
Scale:- 30 fed . = one inch 
Note- VhU Sunny watmada by />i« Tranalx m \d Stadia method lastd on a jyjfn 
ot T>1 angulation and Spirit L*v*U ThtDa | (um plant utaktn\oo net Men 
Nil station Lai at AS' Mt '"■ BnVn or A \agnetic Nfdit tf-*o. 



J, 



APPENDICES. 



APPENDIX A. 



THE JUDICIAL FUNCTIONS OF SURVEYORS. 

BY JUSTICE COOLEY OF THE MICHIGAN SUPREME COURT. 

When a man has had a training in one of the exact sciences, where 
every problem within its purview is supposed to be susceptible of accu- 
rate solution, he is likely to be not a little impatient when he is told that, 
under some circumstances, he must recognize inaccuracies, and govern 
his action by facts which lead him away from the results which theoreti- 
cally he ought to reach. Observation warrants us in saying that this re- 
mark may frequently be made of surveyors. 

In the State of Michigan all our lands are supposed to have been 
surveyed once or more, and permanent monuments fixed to determine 
the boundaries of those who should become proprietors. The United 
States, as original owner, caused them all to be surveyed once by sworn 
officers, and as the plan of subdivision was simple, and was uniform over 
a large extent of territory, there should have been, with due care, few or 
no mistakes ; and long rows of monuments should have been perfect 
guides to the place of any one that chanced to be missing. The truth 
unfortunately is that the lines were very carelessly run, the monuments 
inaccurately placed ; and, as the recorded witnesses to these were many 
times wanting in permanency, it is often the case that when the monument 
was not correctly placed it is impossible to determine by the record, with 
the aid of anything on the ground, where it was located. The incorrect 
record of course becomes worse than useless when the witnesses it refers 
to have disappeared. 

It is, perhaps, generally supposed that our town plats were more ac- 
curately surveyed, as indeed they should have been, for in general there 
can have been no difficulty in making them sufficiently perfect for all 
practical purposes. Many of them, however, were laid out in the woods; 
some of them by proprietors themselves, without either chain or com- 
pass, and some by imperfectly trained surveyors, who, when land was 
cheap, did not appreciate the importance of having correct lines to deter- 
mine boundaries when land should become dear. The fact probably is 
that town surveys are quite as inaccurate as those made under authority 
of the general government. 

It is now upwards of fifty years since a major part of the public sur- 
veys in what is now the State of Michigan were made under authority of 



580 - SURVEYING. 



the United States. Of the lands south of Lansing, it is now forty years 
since the major part were sold and the work of improvement begun. A 
generation has passed away since they were converted into cultivated 
farms, and few if any of the original corner and quarter stakes now re- 
main. 

The corner and quarter stakes were often nothing but green sticks 
driven into the ground. Stones might be put around or over these if 
they were handy, but often they were not, and the witness trees must be 
relied upon after the stake was gone. Too often the first settlers were 
careless in fixing their lines with accuracy while monuments remained, 
and an irregular brush fence, or something equally untrustworthy, may 
have been relied upon to keep in mind where the blazed line once was. 
A fire running through this might sweep it away, and if nothing were sub- 
stituted in its place, the adjoining proprietors might in a few years be 
found disputing over their lines, and perhaps rushing into litigation, as 
soon as they had occasion to cultivate the land along the boundary. 

If now the disputing parties call in a surveyor, it is not likely that any 
one summoned would doubt or question that his duty was to find, if 
possible, the place of the original stakes which determined the boundary 
line between the proprietors. However erroneous may have been the 
original survey, the monuments that were set must nevertheless govern, 
even though the effect be to make one half-quarter section ninety acres 
and the one adjoining but seventy; for parties buy or are supposed to 
buy in reference to those monuments, and are entitled to what is within 
their lines, and no more, be it more or less. Mclver v. Walker,/^ Whea- 
ton's Reports, 444; Land Co. v. Saunders, 103 U. S. Reports, 316; Cot- 
tingham v. Parr, 93 111. Reports. 233 ; Bunion v. Cardivell, 53 Texas Re- 
ports, 408; Watson v. Jones. 85 Penn. Reports, 117. 

While the witness trees remain there can generally be no difficulty in 
determining the locality of the stakes. When the witness trees are 
gone, so that there is no longer record evidence of the monuments, it is 
remarkable how many there are who mistake altogether the duty that 
now devolves upon the surveyor. It is by no means uncommon that we 
find men whose theoretical education is supposed to make them experts 
who think that when the monuments are gone, the only thing to be done 
is to place new monuments where the old ones should have been, and 
where they would have been if placed correctly. This is a serious mis- 
take. The problem is now the same that it was before : to ascertain, by 
the best lights of which the case admits, where the original lines were. 
The mistake above alluded to is supposed to have found expression in 
our legislation ; though it is possible that the real intent of the act to 
which we shall refer is not what is commonly supposed, 

An act passed in 1869, Compiled Laws, § 593, amending the laws re- 
specting the duties and powers of county surveyors, after providing for 
the case of corners which can be identified by the original field-notes or 
other unquestionable testimony, directs as follows : 

" Second. Extinct interior section-corners must be re-established at 
the intersection of two right lines joining the nearest known points on 
the original section lines east and west and north and south of it. 



APPENDIX A. 581 



"Third. Any extinct quarter-section corner, except on fractional lines, 
must be re-established equidistant and in a right line between the section 
corners; in all other cases at its proportionate distance between the 
nearest original corners on the same line." 

The corners thus determined, the surveyors are required to perpetu- 
ate by noting bearing trees when timber is near. 

To estimate properly this legislation, we must start with the admit- 
ted and unquestionable fact that each purchaser from government bought 
such land as was within the original boundaries, and unquestionably 
owned it up to the time when the monuments became extinct. If the 
monument was set for an interior-section corner, but did not happen to 
be " at the intersection of two right lines joining the nearest known 
points on the original section lines east and west and north and south 
of it," it nevertheless determined the extent of his possessions, and he 
gained or lost according as the mistake did or did not favor him. 

It will probably be admitted that no man loses title to his land or any 
part thereof merely because the evidences become lost or uncertain. It 
may become more difficult for him to establish it as against an adverse 
claimant, but theoretically the right remains; and it remains as a poten- 
tial fact so long as he can present better evidence than any other person. 
And it may often happen that, notwithstanding the loss of all trace of a 
section corner or quarter stake, there will still be evidence from which any 
surveyor will be able to determine with almost absolute certainty where 
the original boundary was between the government subdivisions. 

There are two senses in which the word extinct may be used in this 
connection : one the sense of physical disappearance ; the other the 
sense of loss of all reliable evidence. If the statute speaks of extinct 
corners in the former sense, it is plain that a serious mistake was made 
in supposing that surveyors could be clothed with authority to establish 
new corners by an arbitrary rule in such cases. As well might the stat- 
ute declare that if a man lose his deed he shall lose his land altogether. 

But if by extinct corner is meant one in respect to the actual location 
of which all reliable evidence is lost, then the following remarks are per- 
tinent: 

1. There would undoubtedly be a presumption in such a case that 
the corner was correctly fixed by the government surveyor where the 
field -notes indicated it to be. 

2. But this is only a presumption, and may be overcome by any satis- 
factory evidence showing that in fact it was placed elsewhere. 

3. No statute can confer upon a county surveyor the power to "estab- 
lish " corners, and thereby bind the parties concerned. Nor is this a 
question merely of conflict between State and Federal law ; it is a ques- 
tion of property right. The original surveys must govern, and the laws 
under which they were made must govern, because the land was bought 
in reference to them ; and any legislation, whether State or Federal, that 
should have the effect to change these, would be inoperative, because 
disturbing vested rights. 

4. In any case of disputed lines, unless the parties concerned settle 
the controversy by agreement, the determination of it is necessarily a 



582 SURVEYING. 



judicial act, and it must proceed upon evidence, and give full oppor- 
tunity for a hearing. No arbitrary rules of survey or of evidence can 
be laid down whereby it can be adjudged. 

The general duty of a surveyor in such a case is plain enough. He 
is not to assume that a monument is lost until after he has thoroughly 
sifted the evidence and found himself unable to trace it. Even then he 
should hesitate long before doing anything to the disturbance of settled 
possessions. Occupation, especially if long continued, often affords very 
satisfactory evidence of the original boundary when no other is attain- 
able ; and the surveyor should inquire when it originated, how, and why 
the lines were then located as they were, and whether a claim of title 
has always accompanied the possession, and give all the facts due force 
as evidence. Unfortunately, it is known that surveyors sometimes, in 
supposed obedience to the State statute, disregard all evidences of occu- 
pation and claim of title, and plunge whole neighborhoods into quarrels 
and litigation by assuming to " establish " corners at points with which 
the previous occupation cannot harmonize. It is often the case that 
where one or more corners are found to be extinct, all parties concerned 
have acquiesced in lines which were traced by the guidance of some 
other corner or landmark, which may or may not have been trustworthy; 
but to bring these lines into discredit when the people concerned do not 
question them not only breeds trouble in the neighborhood, but it must 
often subject the surveyor himself to annoyance and perhaps discredit, 
since in a legal controversy the law as well as common-sense must declare 
that a supposed boundary line long acquiesced in is better evidence of 
where the real line should be than any survey made after the original 
monuments have disappeared. Stewart vs. Carleton, 31 Mich. Reports, 
270; Diehl vs. Zanger, 39 Mich. Reports, 601 ; Dufiont vs. Starring, 42 
Mich. Reports, 492. And county surveyors, no more than any others, 
can conclude parties by their surveys. 

The mischiefs of overlooking the facts of possession must often appear 
in cities and villages. In towns the block and lot stakes soon disappear; 
there are no witness trees and no monuments to govern except such as 
have been put in their places, or where their places were supposed to be. 
The streets are likely to be soon marked off by fences, and the lots in a 
block will be measured off from these, without looking farther. Now it 
may perhaps be known in a particular case that a certain monument still 
remaining was the starting-point in the original survey of the town plat ; 
or a surveyor settling in the town may take some central point as the 
point of departure in his surveys, and assuming the original plat to be 
accurate, he will then undertake to find all streets and all lots by course 
and distance according to the plat, measuring and estimating from his 
point of departure. This procedure might unsettle every line and every 
monument existing by acquiescence in the town ; it would be very likely 
to change the lines of streets, and raise controversies everywhere. Yet 
this is what is sometimes done ; the surveyor himself being the first 
person to raise the disturbing questions. 

Suppose, for example, a particular village street has been located by 
acquiescence and use for many years, and the proprietors in a certain 



APPENDIX A. 583 



block have laid off their lots in reference to this practical location. 
Two lot-owners quarrel, and one of them calls in a surveyor that he may 
be sure that his neighbor shall not get an inch of land from him. This 
surveyor undertakes to make his survey accurate, whether the original 
was, or not, and the first result is, he notifies the lot-owners that there is 
error in the street line, and that all fences should be moved, say, one foot 
to the east. Perhaps he goes on to drive stakes through the block ac- 
cording to this conclusion. Of course, if he is right in doing this, all 
lines in the village will be unsettled ; but we will limit our attention to 
the single block. It is not likely that the lot-owners generally will allow 
the new survey to unsettle their possessions, but there is always a prob- 
ability of finding some one disposed to do so. We shall then have a 
lawsuit; and with what result? 

It is a common error that lines do not become fixed by acquiescence 
in a less time than twenty years. In fact, by statute, road lines may be- 
come conclusively fixed in ten years ; and there is no particular time 
that shall be required to conclude private owners, where it appears that 
they have accepted a particular line as their boundary, and all concerned 
have cultivated and claimed up to it. McNamara vs. Seaton, 82 111. Re- 
ports, 498; Bunce vs. Bidwell, 43 Mich. Reports, 542. Public policy re- 
quires that such lines be not lightly disturbed, or disturbed at all after 
the lapse of any considerable time. The litigant, therefore, who in such 
a case pins his faith on the surveyor, is likely to suffer for his reliance, 
and the surveyor himself to be mortified by a result that seems to im- 
peach his judgment. 

Of course nothing in what has been said can require a surveyor to 
conceal his own judgment, or to report the facts one way when he be- 
lieves them to be another. He has no right to mislead, and he may 
rightfully express his opinion that an original monument was at one 
place, when at the same time he is satisfied that acquiescence has fixed 
the rights of parties as if it were at another. But he would do mischief 
if he were to attempt to " establish" monuments which he knew would 
tend to disturb settled rights ; the farthest he has a right to go, as an 
officer of the law, is to express his opinion where the monument should 
be, at the same time that he imparts the information to those who em- 
ploy him, and who might otherwise be misled, that the same authority 
that makes him an officer and entrusts him to make surveys, also allows 
parties to settle their own boundary lines, and considers acquiescence in 
a particular line or monument, for any considerable period, as strong, if 
not conclusive, evidence of such settlement. The peace of the com- 
munity absolutely requires this rule. Joyce vs. Williams, 26 Mich. Re- 
ports, 332. It is not long since that, in one of the leading cities of the 
State, an attempt was made to move houses two or three rods into a 
street, on the ground that a survey under which the street had been 
located for many years had been found on more recent survey to be 
erroneous. 

From the foregoing it will appear that the duty of the surveyor where 
boundaries are in dispute must be varied by the circumstances. 1. He 
is to search for original monuments, or for the places where they were 



584 SURVEYING. 



originally located, and allow these to control if he finds them, unless he 
has reason to believe that agreements of the parties, express or implied, 
have rendered them unimportant. By monuments in the case of gov- 
ernment surveys we mean of course the corner and quarter stakes : 
blazed lines or marked trees on the lines are not monuments ; they are 
merely guides or finger-posts, if we may use the expression, to inform us 
with more or less accuracy where the monuments may be found. 2. If 
the original monuments are no longer discoverable, the question of loca- 
tion becomes one of evidence merely. It is merely idle for any State 
statute to direct a surveyor to locate or "establish" a corner, as the place 
of the original monument, according to some inflexible rule. The sur- 
veyor on the other hand must inquire into all the facts ; giving due prom- 
inence to the acts of parties concerned, and always keeping in mind, 
first, that neither his opinion nor his survey can be conclusive upon 
parties concerned ; second, that courts and juries may be required to fol- 
low after the surveyor over the same ground, and that it is exceedingly 
desirable that he govern his action by the same lights and rules that will 
govern theirs. On town plats if a surplus or deficiency appears in a 
block, when the actual boundaries are compared with the original figures, 
and there is no evidence to fix the exact location of the stakes which 
marked the division into lots, the rule of common-sense and of law is 
that the surplus or deficiency is to be apportioned between the lots, on 
an assumption that the error extended alike to all parts of the block. 
O'Brien vs. McGrane, 29 Wis. Reports, 446 ; Quinnin vs. Reixers, 46 
Mich. Reports, 605. 

It is always possible when corners are extinct that the surveyor may 
usefully act as a mediator between parties, and assist in preventing legal 
controversies by settling doubtful lines. Unless he is made for this pur- 
pose an arbitrator by legal submission, the parties, of course, even if they 
consent to follow his judgment, cannot, on the basis of mere consent, be 
compelled to do so; but if he brings about an agreement, and they carry 
it into effect by actually conforming their occupation to his lines, the 
action will conclude them. Of course it is desirable that all such agree- 
ments be reduced to writing; but this is not absolutely indispensable if 
they are carried into effect without. 

Meander Lines. — The subject to which allusion will now be made is 
taken up with some reluctance, because it is believed the general rules 
are familiar. Nevertheless it is often found that surveyors misapprehend 
them, or err in their application; and as other interesting topics are 
somewhat connected with this, a little time devoted to it will probably 
not be altogether lost. The subject is that of meander lines. These 
are lines traced along the shores of lakes, ponds, and considerable rivers 
as the measures of quantity when sections are made fractional by such 
waters. These have determined the price to be paid when government 
lands were bought, and perhaps the impression still lingers in some 
minds that the meander lines are boundary lines, and all in front of 
them remains unsold. Of course this is erroneous. There was never 
any doubt that, except on the large navigable rivers, the boundary of the 
owners of the banks is the middle line of the river; and while some 






APPENDIX A. 585 



courts have held that this was the rule on all fresh-water streams, large 
and small, others have held to the doctrine that the title to the bed of the 
stream below low-water mark is in the State, while conceding to the 
owners of the banks all riparian rights. The practical difference is not 
very important. In this State the rule that the centre line is the bound- 
ary line is applied to all our great rivers, including the Detroit, varied 
somewhat by the circumstance of there being a distinct channel for 
navigation in some cases with the stream in the main shallow, and also 
sometimes by the existence of islands. 

The troublesome questions for surveyors present themselves when the 
boundary line between two contiguous estates is to be continued from the 
meander line to the centre line of the river. Of course the original sur- 
vey supposes that each purchaser of land on the stream has a water-front 
of the length shown by the field-notes ; and it is presumable that he 
bought this particular land because of that fact. In many cases it now 
happens that the meander line is left some distance from the shore by 
the gradual change of course of the stream or diminution of the flow 
of water. Now the dividing line between two government subdivisions 
might strike the meander line at right angles, or obliquely ; and in some 
cases, if it were continued in the same direction to the centre line of the 
river, might cut off from the water one of the subdivisions entirely, or at 
least cut it off from any privilege of navigation, or other valuable use 
of the water, while the other might have a water-front much greater 
than the length of a line crossing it at right angles to its side lines. 
The effect might be that, of two government subdivisions of equal size 
and cost, one would be of very great value as water-front property, and 
the other comparatively valueless. A rule which would produce this re- 
sult would not be just, and it has not been recognized in the law. 

Nevertheless it is not easy to determine what ought to be the correct 
rule for every case. If the river has a straight course, or one nearly so, 
every man's equities will be preserved by this rule : Extend the line of 
division between the two parcels from the meander line to the centre line 
of the river, as nearly as possible at right angles to the general course of 
the river at that point. This will preserve to each man the water front 
which the field-notes indicated, except as changes in the water may have 
affected it, and the only inconvenience will be that the division line be- 
tween different subdivisions is likely to be more or less deflected where it 
strikes the meander line. 

This is the legal rule, and it is not limited to government surveys, but 
applies as well to water lots which appear as such on town plats. Bay 
City Gas Light Co. v. The Industrial Works, 28 Mich. Reports, 182. It 
often happens, therefore, that the lines of city lots bounded on navigable 
streams are deflected as they strike the bank, or the line where the bank 
was when the town was first laid out. 

When the stream is very crooked, and especially if there are short 
bends, so that the foregoing rule is incapable of strict application, it is 
sometimes very difficult to determine what shall be done; and in many 
cases the surveyor may be under the necessity of working out a rule for 
himself. Of course his action cannot be conclusive ; but if he adopts one 



586 



SURVEYING. 



that follows, as nearly as the circumstances will admit, the general rule 
above indicated, so as to divide as near as may be the bed of the stream 
among the adjoining owners in proportion to their lines upon the shore, 
his division, being that of an expert, made upon the ground and with all 
available lights, is likely to be adopted as law for the case. Judicial de- 
cisions, into which the surveyor would find it prudent to look under such 
circumstances, will throw light upon his duties and may constitute a suf- 
ficient guide when peculiar cases arise. Each riparian lot-owner ought to 
have a line on the legal boundary, namely, the centre line of the stream, 
proportioned to the length of his line on the shore ; and the problem in 
each case is, how this is to be given him. Alluvion, when a river imper- 
ceptibly changes its course, will be apportioned by the same rules. 

The existence of islands in a stream, when the middle line constitutes 
a boundary, will not affect the apportionment unless the islands were 
surveyed out as government subdivisions in the original admeasurement. 
Wherever that was the case, the purchaser of the island divides the bed 
of the stream on each side with the owner of the bank, and his rights 
also extend above and below the solid ground, and are limited by the 
peculiarities of the bed and the channel. If an island was not surveyed as 
a government subdivision previous to the sale of the hank, it is of course 
impossible to do this for the purposes of government sale afterwards, for 
the reason that the rights of the bank owners are fixed by their purchase : 
when making that, they have a right to understand that all land between 
the meander lines, not separately surveyed and sold, will pass with the 
shore in the government sale ; and having this right, anything which 
their purchase would include under it cannot afterward be taken from 
them. It is believed, however, that the federal courts would not recog- 
nize the applicability of this rule to large navigable rivers, such as those 
uniting the great lakes. 

On all the little lakes of the State which are mere expansions near 
their mouths of the rivers passing through them — such as the Muskegon, 
Pere Marquette and Manistee — the same rule of bed ownership has been 
judicially applied that is applied to the rivers themselves ; and the divi- 
sion lines are extended under the water in the same way. Rice v. Ruddi- 
?nan, 10 Mich., 125. If such a lake were circular, the lines would con- 
verge to the centre ; if oblong or irregular, there might be a line in the 
middle on which they would terminate, whose course would bear some 
relation to that of the shore. But it can seldom be important to follow 
the division line very far under the water, since all private rights are sub- 
ject to the public rights of navigation and other use, and any private use 
of the lands inconsistent with these would be a nuisance, and punishable 
as such. It is sometimes important, however, to run the lines out for 
some considerable distance, in order to determine where one may law- 
fully moor vessels or rafts, for the winter, or cut ice. The ice crop that 
forms over a man's land of course belongs to him. Lorman v. Benson, 8 
Mich., 18 ; People 's Ice Co. v. Steamer Excelsior, recently decided. 

What is said above will show how unfounded is the notion, which is 
sometimes advanced, that a riparian proprietor on a meandered river may 
lawfully raise the water in the stream without liability to the proprietors 



APPENDIX A. 587 



above, provided he does not raise it so that it overflows the meander line. 
The real fact is that the meander line has nothing to do with such a case, 
and an action will lie whenever he sets back the water upon the proprie- 
tor above, whether the overflow be below the meander lines or above 
them. 

As regards the lakes and ponds of the State, one may easily raise 
questions that it would be impossible for him to settle. Let us suggest 
a few questions, some of which are easily answered, and some not : 

1. To whom belongs the land under these bodies of water, where they 
are not mere expansions of a stream flowing through them ? 

2. What public rights exist in them ? 

3. If there are islands in them which were not surveyed out and sold 
by the United States, can this be done now? 

Others will be suggested by the answers given to these. 

It seems obvious that the rules of private ownership which are applied 
to rivers cannot be applied to the great lakes. Perhaps it should be held 
that the boundary is at low-water mark, but improvements beyond this 
would only become unlawful when they became nuisances, Islands in 
the great lakes would belong to the United States until sold, and might 
be surveyed and measured for sale at any time. The right to take fish in 
the lakes, or to cut ice, is public like the right of navigation, but is to be 
exercised in such manner as not to interfere with the rights of shore 
owners. But so far as these public rights can be the subject of ownership, 
they belong to the State, not to the United States ; and, so it is believed, 
does the bed of a lake also. Pollard v. Hagan, 3 Howard's U. S. Reports. 
But such rights are not generally considered proper subjects of sale, but, 
like the right to make use of the public highways, they are held by the 
State in trust for all the people. 

What is said of the large lakes may perhaps be said also of many of 
the interior lakes of the State ; such, for example, as Houghton, Higgins, 
Cheboygan, Burt's, Mullet, Whitmore, and many others. But there are 
many little lakes or ponds which are gradually disappearing, and the 
shore proprietorship advances pari passu as the waters recede. If these 
are of any considerable size — say, even a mile across — there may be ques- 
tions of conflicting rights which no adjudication hitherto made could 
settle. Let any surveyor, for example, take the case of a pond of irregu- 
lar form, occupying a mile square or more of territory, and undertake to 
determine the rights of the shore proprietors to its bed when it shall 
totally disappear, and he will find he is in the midst of problems such as 
probably he has never grappled with, or reflected upon before. But the 
general rules for the extension of shore lines, which have already been 
laid down, should govern such cases, or at least should serve as guides in 
their settlmeent. Note. — Since this address was delivered some of these 
questions have received the attention of the Supreme Court of Michigan 
in the cases of Richardson v. Prentiss, 48 Mich. Reports, 88, and Backus 
v. Detroit, Albany Law Journal, vol. 26, p. 428. 

Where a pond is so small as to be included within the lines of a pri- 
vate purchase from the government, it is not believed the public have any 
rights in it whatever. VVhere it is not so included, it is believed they have 



533 SURVEYING. 



rights of fishery, rights to take ice and water, and rights of navigation for 
business or pleasure. This is the common belief, and probably the just 
one. Shore rights must not be so exercised as to disturb these, and the 
States may pass all proper laws for their protection. It would be* easy 
with suitable legislation to preserve these little bodies of water as perma- 
nent places of resort for the pleasure and recreation of the people, and 
there ought to be such legislation. 

If the State should be recognized as owner of the beds of these small 
lakes and ponds, it would not be owner for the purpose of selling. It 
would be owner only as a trustee for the public use; and a sale would be 
inconsistent with the right of the bank owners to make use of the water 
in its natural condition in connection with their estates. Some of them 
might be made salable lands by draining ; but the State could not drain, 
even for this purpose, against the will of the shore owners, unless their 
rights were appropriated and paid for. 

Upon many questions that might arise between the State as owner of 
the bed of a little lake and the shore owners, it would be presumptuous to 
express an opinion now, and fortunately the occasion does not require it. 

I have thus indicated a few of the questions with which surveyors may 
now and then have occasion to deal, and to which they should bring good 
sense and sound judgment. Surveyors are not and cannot be judicial 
officers, but in a great many cases they act in a quasi judicial capacity 
with the acquiescence of parties concerned ; and it is important for them 
to know by what rules they are to be guided in the discharge of their 
judicial functions. What I have said cannot contribute much to their 
enlightenment, but I trust will not be wholly without value. 



APPENDIX B. 



INSTRUCTIONS TO U. S. DEPUTY MINERAL SURVEYORS, 
FOR THE DISTRICT OF COLORADO. (1886.) 

GENERAL RULES. 

i. All official communications must be addressed to the Surveyor-Gen- 
eral. You will always refer to the date and subject-matter of the letter 
to which you reply, and when a mineral claim is the subject of corre- 
spondence, you will give the name, ownership and survey number. 

2. You should keep a complete record of each survey made by you, 
and the facts coining to your knowledge at the time, as well as copies of 
all your field-notes, reports and official correspondence, in order that 
such evidence may be readily produced when called for at any future 
time. 

3. Field-notes and other reports must be written in a clear and legible 
hand, and upon the proper blanks furnished by this office. No cut sheets, 
interlineations or erasures will be allowed ; and no abbreviations or sym- 
bols must be used, except such as are indicated in the specimen field- 
notes. 

4. No return by you will be recognized as official unless made in pur- 
suance of a special order from this office. 

5. The claimant is required, in all cases, to make satisfactory arrange- 
ments with you for the payment for your services and those of your 
assistants in making the survey, as the United States will not be held 
responsible for the payment of the same. You will call the attention of 
applicants for mineral-survey orders to the requirements of the circular 
of this date in the appendix. 

6. You will promptly notify this office of any change in your post-office 
address. Upon permanent removal from the State, you are expected to 
resign your appointment. 

NOT TO ACT AS ATTORNEY. 

7. You are precluded from acting, either directly or indirectly, as at- 
torney in mineral claims. Your duty in any particular case ceases when 
you have executed the survey and returned the field-notes and prelimi- 
nary plat, with your report to the Surveyor-General. You will not be al- 
lowed to prepare for the mining claimant the papers in support ol his 



59° SURVEYING. 



application for patent, or otherwise perform the duties of an attorney 
before the land- office in connection with a mining claim. You are not 
permitted to combine the duties of surveyor and notary-public in the 
same case by administering oaths to the parties in interest. In short, 
you must have absolutely nothing to do with the case except in your 
official capacity as surveyor. You will make no survey of a mineral 
claim in which you hold an interest. 

THE FIELD-WORK. 

8. The survey made and reported must, in every case, be an actual sur- 
vey on the ground in full detail, made by you in person after the receipt 
of the order, and without reference to any knowledge you may have pre- 
viously acquired by reason of having made the location-survey or other- 
wise, and must show the actual facts existing at the time. If the season 
of the year, or any other cause, renders such personal examination im- 
possible, you will postpone the survey, and under no circumstances rely 
upon the statements or surveys of other parties, or upon a former exami- 
nation by yourself. 

The term survey in these instructions applies not only to the usual 
field-work, but also to the examinations required for the preparation of 
your affidavits of five hundred dollars expenditure, descriptive reports on 
placer-claims and all other reports. 

SURVEY AND LOCATION. 

9. The survey must be made in strict conformity with, or be embraced 
within, the lines of the record of location upon which the order is based. 
If the survey and location are identical, that fact must be clearly and 
distinctly stated in your field-notes. If not identical, a bearing and dis- 
tance must be given from each established corner of the survey to the 
corresponding corner of the location. The lines of the location, as 
found upon the ground, must be laid down upon the preliminary plat in 
such manner as to contrast and show their relation to the lines of the 
survey. 

10. If the record of location has been made prior to the passage of the 
mining act of May 10, 1872, and is not sufficiently definite and certain to 
enable you to make a correct survey therefrom, you are required, after 
reasonable notice in writing, to be served personally or through the 
United States mail on the applicant for survey and adjoining claimants, 
whose residence or post-office address you may know, or can ascertain by 
the exercise of reasonable diligence, to take testimony of neighboring 
claimants and other persons who are familiar with the boundaries there- 
of as originally located and asserted by the locators of the claim, and 
after having ascertained by such testimony the boundaries as originally 
established, you will make a survey in accordance therewith, and trans- 
mit full and correct returns of the survey, accompanied by the copy of 
the record of location, the testimony, and a copy of the notice served on 
the claimant and adjoining proprietors, certifying thereon when, in what 
manner, and on whom service was made. 

11. If the location has been made subsequent to the passage of the 






APPENDIX B. 591 



mining act of May 10, 1872, and the law has been complied with in the mat- 
ter of marking the location on the ground and recording the same, and 
any question should arise in the execution of the survey as to the iden- 
tity of monuments, marks, or boundaries which cannot be determined by 
a reference to the record, you are required to'take testimony in the man- 
ner hereinbefore prescribed for surveys of claims located prior to May 10, 
1872, and having thus ascertained the true and correct boundaries origi- 
nally established, marked and recorded, you will make the survey accord- 
ingly. 

12. In accordance with the principle that courses and distances must 
give way when in conflict with fixed objects and monuments, you will 
not, under any circumstances, change the corners of the location for the 
purpose of making them conform to the description in the record. If 
the difference from the location be slight, it may be explained in the field- 
notes, but if there should be a wide discrepancy, you will report the facts 
to tffis office and await further instructions. 

INSTRUMENT. 

13. All mineral surveys must be made with a SOLAR transit, or 
other instrument operating independently of the magnetic needle, and 
all courses must be referred to the true meridian. It is deemed best 
that a solar transit should be used under all circumstances. The varia- 
tion should be noted at each corner of the survey. 

CONNECTIONS. 

14. Connect corner No. 1 of your survey by course and distance with 
some corner of the public survey or with a United States location-mon- 
ument, if the claim lies within two miles of such corner or monument. 
If both are within the required distance, you will connect with the near- 
est corner of the public survey. 

LOCATION-MONUMENTS. 

15. In case your survey is situated in a district where there are no 
corners of the public survey and no monuments within the prescribed 
limits, you will proceed to establish a mineral monument, in the location 
of which you will exercise the greatest care to insure permanency as to 
site and construction. 

16. The site, when practicable, should be some prominent point visi- 
ble for a long distance from every direction, and should be so chosen 
that the permanency of the monument will not be endangered by snow, 
rock or land slides, or other natural causes. 

17. The location-monument should consist of a post eight feet long 
and six inches square set three feet in the ground, and protected by a 
well-built conical mound of stone three feet high and six feet base. 
The letters U. S. L. M. followed by a name must be scribed on the post 
and also chiselled on a large stone in the mound, or on the rock in place 
that may form the base of the monument. There is no objection to the 
establishment of a location-monument of larger size, or of other material 
of equally durable character. 

18. From the monument, connections by course and distance must 



59 2 SURVEYING. 



be taken to two or three bearing trees or rocks, and to any well-known 
natural and permanent objects in the vicinity, such as the confluence of 
streams, prominent rocks, buildings, shafts or mouths of adits. Bearings 
should also be taken to prominent mountain-peaks, and the approximate 
distance and direction ascertained from the nearest town or mining 
camp. A detailed description of the location-monument must be in- 
cluded in the field-notes of the survey for which it is established. 

CORNERS. 

19. Corners may consist of 

First — A stone at least twenty-four inches long by six inches square 
set eighteen inches in the ground. 

Second— A post at least four and a half feet long by four inches 
square set twelve inches in the ground and surrounded by a mound of 
stone or earth two and a half feet high and five feet base. 

Third — A rock in place. * 

20. All corners must be established in a permanent and workmanlike 
manner, and the corner and survey number must be neatly chiselled or 
scribed on the sides facing the claim. When a rock in place is used its 
dimensions above ground must be stated, and a cross chiselled at the ex- 
act corner-point. 

21. In case the point for the corner be inaccessible or unsuitable, you 
will establish a witness-corner, which must be marked with the letters 
W. C. in addition to the corner and survey number. The witness-corner 
should be located upon a line of the survey and as near as practicable to 
the true corner, with wmich it must be connected by course and distance. 
The reason for the establishment of a witness-corner must always be 
stated in the field-notes. 

22. The identity of all corners should be perpetuated by taking 
courses and distances to bearing trees, rocks, and other objects, as pre- 
scribed in the establishment of location-monuments. If an official sur- 
vey has been made within a reasonable distance in the vicinity, you will 
run a connecting line to some corner of the same, and connect in like 
manner with all conflicting surveys and claims. 

TOPOGRAPHY. 

23. Note carefully all topographical features of the claim, taking dis- 
tances on your lines to intersections with all streams, gulches, ditches, 
ravines, mountain ridges, roads, trails, etc., with their widths, courses and 
other data that may be required to map them correctly. If the claim 
lies within a town-site, locate all municipal improvements, such as blocks, 
streets and buildings. 

24. You are required also to locate all mining and other improve- 
ments upon the claim by courses and distances from corners of the sur- 
vey, or by rectangular offsets from the centre line, specifying the dimen- 
sions and character of each in full detail. 

CONFLICTS. 

25. If in running the exterior boundaries of a claim, you find that two 
surveys conflict, you will determine the courses and distances from the 



APPENDIX B. 593 



established corners at which the exterior boundaries of the respective 
surveys intersect each other, and run all lines necessary for the determi- 
nation of the areas in conflict, both with surveyed and unsurveyed 
claims. You are not required, however, to show conflicts with unsur- 
veyed claims unless the same are to be excluded. 

26. When a placer-claim includes lodes, or when several lode-loca- 
tions are included as one claim in one survey, you will preserve a con- 
secutive series of numbers for the corners of the whole survey in each 
case, In the former case you will first describe the placer-claim in your 
field-notes. 

PLACER-CLAIMS AND MILL-SITES. 

27. The exterior lines of placer-claims cannot be extended over other 
claims, and the conflicting areas excluded as with lode-claims, it being 
the surface ground only, w T ith side lines taken perpendicularly downward 
for which application is made. The survey must accurately define the 
boundaries of the claim. The same rule will apply to the survey of mill- 
sites. 

28. If by reason of intervening surveys or claims a placer or mill-site 
survey should be divided into separate tracts, you will also preserve a 
consecutive series of numbers for the corners of the whole survey, and 
distinguish the detached portions as Lot No. 1, Lot No. 2, etc., connect- 
ing by course and distance a corner of each lot with some corner of the 
one previously described. 

LODE AND MILL-SITE. 

29. A lode and mill-site claim in one survey will be distinguished by 
the letters A and B following the number of the survey. The corners of 
the mill-site will be numbered independently of those of the lode. Cor- 
ner No. 1 of the mill-site must be connected with a corner of the lode 
claim as well as with a corner of the public survey or U. S. location- 
monument. 

FIELD-NOTES. 

30. In order that the results of your survey may be reported to this 
office in a uniform manner, you will prepare your field-notes and pre- 
liminary plat in strict conformity with the specimen field-notes and plat, 
which are made part of these instructions. They are designed to furnish 
vou with all needed information concerning the manner of describing 
the boundaries, corners, connections, intersections, conflicts and improve- 
ments, and stating the variation, area, location and other data con- 
nected with the survey of mineral claims, and contain forms of affidavits 
for the deputy-surveyor and his assistants. 

In your first reference to any other mineral claim you will give the 
name, ownership, and if surveyed, the survey-number. 

31. The total area of a lode-claim embraced by the exterior bounda- 
ries, and also the area in conflict with each intersecting survey or claim 
should be so stated, that the conflicts with any one or all of them may 
be included or excluded from your survey. This will enable the claim- 
ant to state in his application for patent the portions to be excluded in 
express terms, and to readily determine the net area of his claim. 



594 SURVEYING. 



32. You will state particularly whether the claim is upon surveyed or 
unsurveyed public lands, giving in the former case the quarter-section, 
township and range in which it is located, and in the latter the township, 
as near as can be determined. 

33. The field-notes must contain the post-office address of the claim- 
ant or his authorized agent. 

EXPENDITURE OF FIVE HUNDRED DOLLARS. 

34. The claimant is required by law, either at the time of filing his 
application, or at any time thereafter, within the sixty days of publica- 
tion, to file with the Register the certificate of the Surveyor-General that 
five hundred dollars' worth of labor has been expended or improvements 
made upon the claim by himself or grantors. The information upon 
which to base this certificate must be derived from the deputy who 
makes the actual survey and examination upon the premises, and such 
deputy is required to specify with particularity and full detail the char- 
acter and extent of such improvements. See also Sec. 8. 

35. When a survey embraces several locations or claims held in coi7i- 
mon, constituting one entire claim, whether lode or placer, an expendi- 
ture of five hundred dollars upon such entire claim embraced in the sur- 
vey will be sufficient and need not be shown upon each of the locations 
included therein. 

36. In case of a lode and mill-site claim in the same survey, an ex- 
penditure of five hundred dollars must be shown upon the lode-claim 
only. 

37. Only actual expenditures and mining improvements, made by the 
claimant or his grantors, having a direct relation to the development of 
the claim, can be included in your estimate. 

38. The expenditures required may be made from the surface, or in 
running a tunnel for the development of the claim. Improvements of' 
any other character, such as buildings, machinery or roadways, must be 
excluded from your estimate unless you show clearly that they are asso- 
ciated with actual excavations, such as cuts, tunnels, shafts, etc., and are 
essential to the practical development of the survey-claim. 

39. You will give in detail the value of each mining improvement in- 
cluded in your estimate of expenditure, and when a tunnel or other 
improvement has been made for the development of other claims in con- 
nection with the one for which survey is made, your report must give the 
name, ownership and survey-number, if any, of each claim to which a 
proportion or interest is credited, and the value of the proportion or 
interest credited to each. The value of improvements made upon other 
locations or by a former locator who has abandoned his claim cannot be 
included in your estimate. 

40. In making out your certificate of the value of the improvements, 
you will follow the form prescribed in the specimen field-notes. 

41. Following your certificate you will locate and describe all other 
improvements made by the claimant or other parties within the bounda- 
ries of the survey. 

42. If the value of the labor and improvements upon a mineral claim 






APPENDIX B. 595 



is less than five hundred dollars at the time of survey, you are authorized 
to file your affidavit of five hundred dollars expenditure at any time 
before the expiration of the sixty days of publication, but not afterwards 
unless by special instructions. 

DESCRIPTIVE REPORTS ON PLACER-CLAIMS. 

43. By General Land Office circular, approved September 23, 1882, you 
are required to make a full examination of all placer-claims at the time 
of survey, and file with your field-notes a descriptive report in which you 
will describe — 

(a) The quality and composition of the soil, and the kind and amount 
of timber, and other vegetation. 

{b) The locus and size of streams, and such other matters as may ap- 
pear upon the surface of the claims. 

(c) The character and extent of all surface and underground workings, 
whether placer or lode, for mining purposes. 

(d) The proximity of centres of trade or residence. 

(e) The proximity of well-known systems of lode deposits or of indi- 
vidual lodes. 

(f) The use or adaptability of the claim for placer-mining, and 
whether water has been brought upon it in sufficient quantity to mine 
the same, or whether it can be procured for that purpose. 

(g) What works or expenditures have been made by the claimant or 
his grantors for the development of the claim, and their situation and 
location with respect to the same as applied for. 

(Ji) The true situation of all mines, salt-licks, salt-springs, and mill- 
seats, which come to your knowledge, or report that none exist on the 
claim, as the facts may warrant. 

(/) Said report must be made under oath, and duly corroborated by 
one or more disinterested persons. 

44. Descriptive reports upon placer-claims taken by legal subdivisions 
are authorized only by special order, and must contain a description of 
the claim in addition to the foregoing requirements. 

PRELIMINARY PLAT. 

45. You will file with your field-notes a preliminary plat on drawing- 
paper or tracing-muslin, protracted on a scale of two hundred feet to an 
inch, on which you will note accurately all the topographical features 
and details of the survey in conformity with the specimen plat herewith. 
Pencil sketches will not be accepted. 

REPORT. 

46. You will also submit with your return of survey a report upon the 
following matters incident to the survey, but not required to be embraced 
in the field-notes. 

47. If the meridian from which your courses were deflected was estab- 
lished by other means than by the solar apparatus attached to your 
transit, you will state in detail your observations and calculations for the 
establishment of such meridian. 



Sg6 SURVEYING. 



48. If any of the lines of the survey were determined by triangulation 
or traverse, you will give in full detail the calculations whereby you ar- 
rived at the results reported in your field-notes. You will also submit 
your calculations of areas of placer and mill-site claims or other irregular 
tracts. 

49. You will mention in your report the discovery of any material 
errors in prior official surveys, giving the extent of the same. 

ERRORS. 

50. Whenever a survey has been reported in error, the deputy-sur- 
veyor who made it will be required to promptly make a thorough exami- 
nation, upon the premises, and report the result under oath to this office. 
In case he finds his survey in error, he will report in detail all discrep- 
ancies with the original survey, and submit any explanation he may have 
to offer as to the cause. If, on the contrary, he should report his survey 
correct, a joint survey will be ordered to settle the differences with the 
surveyor who reported the error. 

JOINT SURVEY. 

51. A joint survey must be made within ten days after the date of 
order, unless satisfactory reasons are submitted, under oath, for a post- 
ponement. 

52. The field-work must in every sense of the term be a joint and not 
a separate survey, and the observations and measurements taken with the 
same instrument and chain, previously tested and agreed upon. 

53. The deputy-surveyor found in error, or if both are in error, the 
one who reported the same will make out the field-notes of the joint sur- 
vey, which, after being duly signed and sworn to by both parties, must 
be transmitted to this office. 

54. The surveyor found in error will be required to pay all expenses 
of the joint survey and preliminary examinations incident thereto, includ- 
ing ten dollars per day to the surveyor whose work is proved to be sub- 
stantially correct. 

55. Your field-work must be accurately and properly performed, and 
your returns made in conformity with the foregoing instructions. Errors 
in the survey must be corrected, at your own expense, and if the time re- 
quired in the examination of your returns is increased by reason of your 
neglect or carelessness you will be required to make an additional deposit 
for office work. You will be held to a strict accountability for the faith- 
ful discharge of your duties, and will be required to observe fully the re- 
quirements and regulations in force as to making mineral surveys. If 
found incompetent as a surveyor, careless in the discharge of your duties, 
or guilty of a violation of said regulations, your appointment will be 
promptly revoked. 

56. All former instructions inconsistent with the foregoing are hereby 
recalled. 



APPENDIX. 



597 



SPECIMEN PRELIMINARY PLAT. 



SURVEY N0.4225 A & B 
DISTRICT N0.3. 



R.81 W.. 




SPECIMEN FIELD-NOTES. 



Survey No. 4225 A and B. 
District No. 3. 

FIELD-NOTES. 

Of the survey of the claim of The Argentum Mining Company, upon 
the Silver King and Gold Queen lodes, and Silver King Mill site, in 
Alpine Mining District, Lake county, Colorado. 

Surveyed by George Lightfoot, U. S. Deputy Mineral Surveyor. 
Survey begun April 22d, 1886, and completed Apr^l 24th, 1886. 

Address of claimant, Wabasso, Colorado. 



--j-.,. 



FEET. 



1242. 
I365.28 



152. 
300. 



SURVEY NO. 4225 A. 

SILVER KING LODE. 

Beginning at Cor. No. 1. 

Identical with Cor. No. 1 of the location. 

A spruce post, 5 ft. long, 4 ins. square, set 2 ft. in the 
ground, with mound of stone, marked 42 1 25 A, whence 

The W. \ cor. Sec. 22, T. 11 S., R. 81 W. of the 6th Prin- 
cipal Meridian, bears S. 79 34' W. 1378.2 ft. 

Cor. No. 1, Gottenburg lode (un surveyed), Neals Mattson, 
claimant, bears S. 40 29' W. 187.67 ft. 

A pine 12 ins. dia., blazed and marked B. T. 4 2 h 5 A, hears 
S. 7 25' E. 22 ft. 

Mount Ouray bears N. n° E. 

Hiawatha Peak bears N. 47 45' W. 

Thence S. 24 45' W. 



Va. 



15 12' E. 



To trail, course N. W. and S. E. 
To Cor. No. 2. 

A granite stone 25x9x6 ins., set 18 ins. in the ground, 
chiselled -^s A, whence 

Cor. No. 2 of the location bears S. 24 45' W. 134.72 ft. 
Cor. No. 1, Sur. No. 2560, Carnarvon lode, David Davies et 
al., claimants bears S. 3 28' E. 116. 6 ft. 

North end of bridge over Columbine creek bears S. 65 15' 
E. 650 ft. 

Thence N.65 15' W. 

Va. 1 5 20' E. 

Intersect line 4-1, Sur. No. 2560, at N. 38 52' W., 231.2 ft. 

from Cor. No. 1. 
To Cor. No. 3. 



APPENDIX B. 



599 



FEET. 



734 

150. 

237. 
IOOO.9 

1365.28 



28.5 

65. 
300. 



285. 
315. 

688.3 

1438. 
1500. 



A cross at corner-point, and ^ T A chiselled on a granite 
rock in place, 2ox 14x6 ft. above the general level, whence 
Cor. No. 3 of the location bears S. 24 45' W. 13472 ft. 
A spruce 16 ins. dia., blazed and marked B. T. ¥ ^ T A, 
bears S. 58 W. 18 ft. 

Thence N. 24 45' E. 

Va. 1 5 20' E. 

Intersect line 4-1 Sur. No. 2560 at N. 38 52' W. 396.4 ft. 

from Cor. No. 1. 
Intersect line 6-7 of this survey. 
To trail, course N. W. and S. E. 
Intersect line 2-3, Gottenburg lode, at N. 25 56' W. 76.26 ft. 

from Cor. No. 2. 
To Cor. No. 4. 

Identical with Cor. No. 4 of the location. 
A pine post 4. 5 ft. long 5 ins. square, set one foot in the 
ground, with mound of earth and stone, marked 
whence 

A cross chiselled on rock in place, marked B. K. ^f^ 
bears N. 28 10' E. 58.9 ft. 

Thence S. 65 15' E. 

Va. 1 5 12' E. 

Intersect line 4-1, Gottenburg lode, at N. 25° 56' W 285.15 ft. 

from Cor. No. 1. 
Intersect line 5-6 of this survey. 
To Cor. No. 1, the place of beginning. 



A, 



GOLD QUEEN LODE. 

Beginning at Cor. No. 5, 

A pine post 5 ft. long, 5 ins. square, set 2 ft. in the ground, 
with mound of earth and stone, marked ^Vs A ' whence 

Cor. No. 1 of this survey bears S. 14° 54' E. 370.16 ft. 

A pine 18 in. dia. bears S. 33 15' W. 51 ft., and a silver 
spruce 13 ins. dia. bears N. 6o° W. 23 ft., both blazed and 
marked B. T. ^fa A. 

Thence S. 24 30' W. 
Va. 1 5 14' E. 
Intersect line 4-1 of this survey. 
Intersect line 4-1, Gottenburg lode, at N. 25 56' W. 237.78 ft. 

from Cor. No. 1. 
Intersect line 1-2, Gottenburg lode, at N. 64 04 E. 12.23 ft. 

from Cor. No. 2. 
To trail, course N. W. and S. E. 
To cor. No. 6, . 

A granite stone 34 x 14x6 ins., set one foot in the ground 
to bedrock, with mound of stone, chiselled 3T Vtt A, whence 

A cross chiselled on ledge of rock marked B. R. ^Vr A > 
bears due north 12 ft. 



6oo 



SURVEYING. 



FEET. 

7o.3 
223.37 

300. 



38.43 

165. 
I04373 

1432.90 
1500. 



300. 



Thence N. 65 30' W. 
Va. i5°2o ; E. 
Intersect line 3-4 of this survey. 
Intersect line 4-1, Sur. No. 2560 at N. 38 52' W 167 28 ft 

from Cor. No. 1. 
To Cor. No. 7. 

A cross at corner-point and^ A chislled on a granite 
boulder 12 x 6 x 3 ft. above ground, whence 

A cross chiselled on vertical face of cliff, marked B R 
4^3- A > bears N. 72 W. 56.2 ft. 

N. m° P E ne 3 9 4 ft nS ' ^ blaZCd and marked B * T - '^ A ' bears 
Thence N 24 30' E. 
Va. not determined on account of local attraction 
Intersect line 4-1, Sur. No. 2560, at N. 38 52' W. 653 ft.' from 

To trail, course N. W. and S. E. 

Intersect line 2-3, Gottenburg lode, at N. 25 56' W 170 06 ft 
from Cor. No. 2. j j -osy * 

Intersect line 4-1, Gottenburg lode, at N. 25 56' W 626 qa ft 
from Cor. No. 1. •>** »•• 

To Cor. No. 8. 

A spruce post 6 ft. long, 5 ins. square, set 2.5 ft. in the 
ground with mound of stone, marked ^J^ A whence 

beafs S rc 9 S ^? E H i 5.8°?t. r ° Ck *" place > marke ' d B^^A, 

N. 8 A 3 oP E e 28 20 5 ft?' ^^ ^^^ and markGd B - T * *** A ' bears 
Thence S. 65 30' E. 
Va. i5°i6'E. 
10 Cor. No. 5, the place of beginning. 
Area. 

Total area of Silver King lode n 40 . acres 

Less area in conflict with 

Sur. No. 2560 124 acre 

Gottenburg lode 1.363 « 1.487 acres 

Net area of Silver King lode 7.916 acres 



Total area of Gold Queen lode 10 **i 

Area in conflict with " 

Sur. No. 2560 034 

Gottenburg lode 2.679 

Silver King lode . . .. i.ggj 

rr . S , ilver Kin g lode (exclusive of conflict of 
said Silver King lode with the Gottenburg lode) 1.309 



acres 



APPENDIX B. 



60 1 



FEET. 



90. 
208. 
504.8 



35i- 
3944 



15. 

40. 



37o- 
647.2 



Total area of Gold Queen lode 10.331 acres 

Less area in conflict with 

Sur. No. 2560 034 acre 

Gottenburg lode 2.679 " 

Silver King lode i-3°9 " 4.022 acres 

Net area of Gold Queen lode 6. 309 acres 

" " Silver King lode 7.916 " 

Net area of lode claim ... 14.225 acres 

SURVEY NO. 4225 B. 

SILVER KING MILL-SITE. 

Beginning at Cor. No. 1, 

A gneiss stone 32x8x6 ins., set 2 ft. in the ground, chis- 
elled 42W B, whence W.£ cor. Sec. 22, T. 11 S, R. 81 W. of 
the 6th Principal Meridian, bears N. 8o° W. 1880 ft. 

Cor. No. 1, Sur. No. 4225 A, bears N. 40 44' W. 760.2 ft. 

A cottonwood 18 ins. dia., blazed and marked ^Vr B, 
bears S. 5 30' E. 17 ft. 

Thence S. 34 E. 
Road to Wabasso, course N. E. and S. W. 
Right bank of Columbine creek, 75 ft. wide, flows S. W. 
To Cor. No. 2, 

An iron bolt 18 ins. long, 1 in. dia., set one foot in rock in 
place, chiselled 4-^-5- B, whence 

A cottonwood, blazed and marked B. T. -^^ B, bears E. 
182 ft. 

Thence S. 56 W. 
Left bank of Columbine creek. 
To Cor. No. 3, 

A point in bed of creek, unsuitable for the establishrne«t 
of a permanent corner. 

Thence N. 34 W. 
Right bank of Columbine creek. 
To witness-corner to Cor. No. 3, 

A pine post 4.5 ft. long, 5 ins. square, set one foot in 
ground, with mound of stone, marked W. C. 4% 2 ? B, whence 

A cottonwood 15 ins. dia. bears N. n° E. 16.5 ft. and a 
cottonwood 19 ins. dia. bears N. 83 W. 23 ft., both blazed 
and marked B. T. W. C. ^- 2J B. 
Road to Wabasso, course N. E. and S. W. 
To Cor. No. 4, 

A gneiss stone 24 x 10x4 ins., set 18 ins. in the ground, 
chiselled 42 4 2 g B, whence 

A cross chiselled on ledge of rock, marked B. R. 4-5V5 B, 
bears N. 85 10' E. 26.4 ft. 

Thence N. 48°43'E. 



602 



SUJ? VE YING. 



FEET. 
125.5 



158.3 



270. 



To Cor. No. 5, 

A gneiss stone 30x8x5 ins., set 2 ft. in the ground, 
chiselled 4^3-5 B. 

Thence S. 34 E. 
To Cor. No. 6, 

A pine post 5 ft. long, 5 ins. square, set 2 ft. in the ground 
with mound of earth and stone, marked 42 6 2g B, whence 

A pine 12 ins. dia., blazed and marked B. T. -±§$-5 B, bears 
S. 33 E. 63.5 ft. 

Thence N. 56 E. 
To Cor. No. 1, the place of beginning, 
Containing 5 acres. 

Variation at all the corners, 15 20' E. 



The surveys of the Gold Queen lode and Silver King mill 
site are identical with the respective locations. 



LOCATION. 

This claim is located in the W. | Sec. 22, T. 1 1 S., R. 81 W. 



EXPENDITURE OF FIVE HUNDRED DOLLARS. 

I certify that the value of the labor and improvements 
upon this claim, placed thereon by the claimant and its 
grantors, is not less than five hundred dollars, and that said 
improvements consist of 

The discovery shaft of the Silver King lode, 6x3 ft., 10 
ft. deep in earth and rock, which bears from Cor. No. 2 N. 6° 
42' W. 287.5 ft. Value $80. 

An incline 7x5 ft., 45 ft. deep in coarse gravel and rock, 
timbered, course N. 58 15' W., dip 62 , the mouth of which 
bears from Cor. No. 2 N. 15 37' E. 908 ft. Value $550. 

The discovery shaft of the Gold Queen lode. 5x5 ft., 18 
ft. deep in rock, which bears from Cor. No. 7 N. 67 39' E. 
219.3 ft., at the bottom of which is a cross-cut 6.5x4 ft. 
running N. 59 26' W. 75 ft. 

Value of shaft and cross-cut, $1,000. 

A log shaft-house 14 ft. square, over the last-mentioned 
shaft. Value $100. 

Two-thirds interest in an adit 6.5 x 5 ft., running due west 
835 ft., timbered, the mouth of which bears from Cor. No. 2 
N. 6i° 15' E. 920 ft. 

This adit is in course of construction for the development 
of the Silver King and Gold Queen lodes of this claim, and 
Sur. No. 2560, Carnarvon lode, David Davies et al., claimants, 
the remaining one-third interest therein having already been 
included in the estimate of five hundred dollars expenditure 
upon the latter claim, Total value of adit, $13,000. 



APPENDIX B. 



603 



A drift 6.5 x4 ft. on the Silver King lode, beginning at a 
point in adit 800 ft. from the mouth, and running N. 20 20' 
E. 195 ft., thence N. 54 15' E. 40 ft. to breast. Value $2,800. 

I further certify that no portion of the improvements 
claimed have been included in the estimate of five hundred 
dollars expenditure upon any other claim. 



OTHER IMPROVEMENTS. 

A log cabin 35x28 ft., the S. W. corner of which bears 
from Cor. No. 7 N. 30 44' E. 496 ft. 

A dam 4 ft. high, 50 ft. long, across Columbine creek, the 
south end of which bears from Cor. No. 2 of the mill-site N. 
58 20' W. 240 ft. 

Said cabin and dam belong to The Argentum Mining 
Company. 

An adit 6x4 ft., running N. 70 50' W. 100 ft., the mouth 
of which bears from Cor. No. 5 S. 58 12' W. 323 ft. belong- 
ing to Neals Mattson, claimant of the Gottenburg lode. 



INSTRUMENT. 

The survey was made with a Young & Sons mountain 
transit No. 5322, with Smith's solar attachment. The courses 
were deflected from the true meridian as determined by solar 
observations. The distances were measured with a 50 ft. 
steel tape. 

employe's certificate. 

List of the names of individuals employed to assist in running, meas- 
uring and marking the lines and corners described in the foregoing field- 
notes of the survey of the claim of The Argentum Mining Company 
upon the Silver King and Gold Queen lodes and Silver King mill-site, in 
Alpine Mining District, Lake County, Colorado. 

William Sharp, 
Robert Talc. 
We hereby certify that we assisted George Lightfoot U. S. Deputy 
Mineral Surveyor, in surveying the exterior boundaries and marking the 
corners of the claim of The Argentum Mining Company upon the Silver 
King and Gold Queen lodes and Silver King mill-site in Alpine Mining 
District, Lake County, Colorado, and that said survey has been in all re- 
spects, to the best of our knowledge and belief, well and faithfully sur- 
veyed and the boundary monuments planted according to the instruc- 
tions furnished by the Surveyor-General. 

William Sharp, 
Robert Talc. 
Subscribed and sworn to by the above-named persons before me, this 
26th day of April, 1886. 

[Seal] John Doolittle, 

Notary Public. 



604 SURVEYING. 



surveyor's oath. 

I, George Lightfoot, U. S. Deputy Mineral Surveyor, do solemnly 
swear that in pursuance of an order from Jas. A. Dawson, Surveyor-Gen- 
eral of the public lands in the State of Colorado, bearing date the 30th 
day of March 1886, and in strict conformity with the laws of the United 
States, and instructions furnished by said Surveyor-General, I have 
faithfully surveyed the claim of The Argentum Mining Company upon 
the Silver King and Gold Queen lodes and Silver King mill-site in Alpine 
Mining District, Lake County, Colorado, and do further solemnly swear 
that the foregoing are the true and original field-notes of such survey, 
and that the improvements are as therein stated. 

George Lightfoot, 
U. S. Deputy Mineral Surveyor. 

Subscribed by said George Lightfoot, U. S. Deputy Mineral Surveyor, 
and sworn to before me this 26th day of April, 1886. 

[Seal] John Doolittle, 

Notary Public. 



APPENDIX C. 



FINITE DIFFERENCES. 



THE CONSTRUCTION OF TABLES. 



In the accompanying figure the ordinates are spaced at the uniform 
distance / apart. Let the successive values of these ordinates, and their 
several orders of differences, be represented by the following notation : 




I 



I I I 

Fig. 152. 

Values of the function, ti , hi, hi, h 3 , hi, h 5 , h 6 . 

First order of differences, A'^ , A'b v A' h v A'/ l3 , A' h v A'/ iy 

Second " " A' 'a . A"k v A"a 2 , A"; lz , A"h v 

Third " " J'"*,, A'"a v A'" hv A'"k v 

Fourth " V A iv / l0 , A w a v A lr k r 

etc., etc. 



606 SUR VE YING. 



We may now write 

hi = k\ -j- ^'a ; 

£ 2 = h x + ^ = /^o + A' h > + ^ + A" K = h + 2^' Ao + ^%; 
1 ^ 3 = A 2 + ^'a 2 = ^o + 3^%> + 3^"^ + A "'h*\ 
h 4 = ho + 4^0 + 6 ^"^o + 4^'"^ + AU K\ 



A! i nin — i) ... . « (w — i) (ii — 2) .... 
h n = *o + «^'*„ + "V-T Jj *o H T ' - A " A. + etc. 



(I) 



It is to be observed that the coefficients follow the law of the bino- 
mial development. It is also seen that the first of the successive orders 
of differences are alone sufficient to enable any term of the function to 
be computed. We will now proceed to find these first terms of the 
several orders of differences for any given equation. 

Almost all functions of a single variable can be developed by the aid 
of Maclaurin's Formula, in the form 

y = Co + £i*o + CVtfo 2 -f" C 3 x s -j- C 4 x 4 -f- etc (2) 

If x take an increment A x , thus becoming Xi, the change in y will 
be represented by A'y and its value will be the new value of the function 
minus its initial value, or A f y Q =yi — jKo. By putting x + A x for x in the 
above equation, developing, subtracting the original equation, and re- 
ducing, we would obtain 

y Y — yo = A'y — (Ci + 2C-2X0 + SCVro 2 + 4CVz"o 3 )^.r 

+ (C 2 + 3CV0 + tCixf)A* x + (C 3 + ^xo)A* x + C^ x , . (3) 

assuming that the function stops with dx *. 

If xi should now take another increment A Xti equal to the previous 
one, we would have X* = Xi + A x and jj/ 2 = yi + A 'y x . Now ^'^ is the 
value A' y when x has become #1, and the difference between A'y Q and 
A'y x is the change in the value of ^/ j due to this change in x. 

Hence A' yi — A'y — A"y . 

To find the value of A" y ^ substitute x + A x for x in equation (3), 
develop, subtract equation (3), reduce, and obtain 

A"y = ( 2 C 2 + 6C s x + i2C 4 *oV 2 .* + ( 6< ^ + 24C*4^oM 3 ^ + hCa^x. (4) 

Similarly we find 

A'"y = {6C 3 -{-24C i xo)A^ x -\-36C i A^ (5) 

A'"y = 2<\C i A i x (a constant). . . (6) 



APPENDIX C. 



607 



From the above development we see — 

1. That the number of orders of differences is equal to the highest ex- 
ponent of the variable involved, the last difference being a constant. 

2. That if any initial value (x ) of the variable be taken, the first of 
the several orders of differences can be obtained in terms of this initial 
value, its constant increment, and the constant coefficients. This fur- 
nishes a ready means of computing a table of values of the function, if 
it can be represented in the form of equation (1). Evidently if the ini- 
tial value of the variable (x ) be taken as zero, the evaluation for the 
several initial differences is much simplified, for then all the terms in x 
disappear. If the constant increment be also taken as unity, the labor 
is still further reduced. 



Example. — Construct a table of values of the function 
y = 50 — 40X -f- 20.* 2 -f- 4x' s — x 4 . 



(7) 

Let the initial value of the variable be zero and the increments unity. 
Evaluating the initial differences by equations (3) to (6), we find, for^o = o, 
and A x — 1, 

yo = + 50; 

A' yo2 = d + C 2 + C 3 + C 4 = - 17; 
A" yo = 2C2 + 6C 3 -f 14C4 = + 50; 
A'" y * - 6C 3 + 36C4 = - 12; 

A [y yO = 24 C 4 = — 24. 

From these initial values we may readily construct the following 
table : 



Values of 


Values of 


1st Differences. 


2d Differences. 


3d Differences. 


4th Differences. 


X. 


y '. ^ 


A' y . 


A "y 


A'"y. 


A'V 


O 


50 


— 17 








I 


33 


+ 33 


+ 50 


— 12 




2 


66 


+ 7i 


+ 38 


" 36 


- 24 


3 


137 


+ 73 


+ 2 


— 60 


- 24 


4 


210 


+ 15 


- 58 


- 84 


- 24 


5 


225 


— 127 


— I42 


— IO8 


- 24 


6 


9 3 


— 377 


— 250 


— 132 


- 24 


7 


— 279 


— 759 


- 382 


etc. 


etc. 


8 


- 1038 


etc. 


etc. 






etc. 


etc. 











* Fig. 152 is the locus of this curve, the ordinates being taken from this 
column. 



6o8 SUR VE YING. 



The initial values in all the columns being given, the table is made 
by continual additions, one column after another, working from right to 
left. Thus, the 4th difference being constant, the initial value, —24, is 
simply repeated indefinitely. The column of 3d differences is now com- 
puted by adding continuously —24 to the preceding value. The column 
of 2d differences is next made out, the quantity to be added each time 
being the intervening 3d difference, which is not constant. In a similar 
manner proceed with the column of 1st differences, and finally with the 
values of the function itself. 

The above formulae apply to all functions of a single variable not 
higher than the fourth degree. Evidently any of the C coefficients may 
be zero, and so cause one or more of the powers of x to entirely disappear. 
If the variable is involved to a higher degree than the fourth, a new de- 
velopment may be made, or the initial values of the successive orders of 
differences may be determined by simply evaluating the function for a 
series of successive values of the variable, one more in number than the 
degree of the equation, and then working out the successive columns of 
differences from these until the last, or constant, difference is found. 
The table may then be continued by combining these differences, as be- 
fore. Thus in the above example the first five values of y might have 
been found by direct evaluation of the function for the corresponding 
values of x, and then the successive differences taken out until the con- 
stant fourth difference, — 24, was found. This can always be done with* 
out resorting to any algebraic discussion as given above. 



THE EVALUATION OF IRREGULAR AREAS. 

The ordinates to any curve, as that in Fig. 152 for instance, may be 
represented by such an equation as the last of equations, (1), where the 
length of any ordinate is given in terms of its number from the initial or- 
dinate, the value of this first ordinate, and the first of the successive 
orders of differences. This equation is 

z. 1 l A', 1 "(* — *) si», , n(n—i)(n — 2) A ,„ t 

h n = h* + nA h Q -\ Y~2 °^ 172~.~3 ^o + etc -» 

where h n is the nth., and therefore any ordinate to the curve. The con- 
stant distance between the ordinates apparently does not enter the equa- 
tion, but it is really represented in the several A's. 

By the calculus the area of any figure included between any curve, 

the axis of abscissas, and two extreme ordinates is A = I hdx, where h 

is the general value of an ordinate, = h n in the above equation, where it 
is shown to be a function of n. Also x = nl where / is the constant 
distance between ordinates, whence dx = Idn. Substituting these val- 
ues of h and dx, we have 



APPENDIX C. 



609 



A = 1 hdx = I h n ldn = I / h n dn = / h Q I dn-\-A'/ l(l j ndn 

+ 7— -. / «(« - iK» + 7^r~7 / w ( w " x ) ( w "" 2 ^ w 
1 J «/o 1.2. 3^/ 



H — 7~ J n(n — 1) (n — 2) (» — 3) </« -j- etc. , 



(8) 



Integrating this equation, we obtain 

2 v o 4 ' x 24 ' 

+ (J5! - "A + ^ - t)^^.+ (^ - f 6 + ^- - ^ + *) ^ £ 

M2O ID 72 8 < x 720 OO QO 36 IO / 

+ (i - J* U EZ!L § - g + ^_ 3 _ n 2 )j % + etc> 
^5040 288 720 04 1080 12 y 



,4 = / 



(9) 



From the schedule of differences on p. 605 we may at once find the 
initial values of the several orders of differences in terms of the succes- 
sive values of the function. Thus 

A'h a — h x — h ; 

A"k = A'h x — A'h<> = hi — 2hi + ho\ 

A'"t = 4"k x - A"k = A'h t — 2A' hl -f Ah* - hz — 3^ 2 -f -$h x - A ; 

A' l A A = A'"k x - A'"/ l0 = J" Aa - iA" hx + A" ho = A' h3 - 3 A' Ai -f 3 A' Al -A'/ io 
= hi — 4/^3 + 6^3 — 4^1 -j- k . 

Again, the coefficients follow the law of the binomial development, 
and we may write 



A t r. 1 n ( H ~ J )r 

A"k = h n - nh n - 1 H — —r—hn - ■. 



I . 2 



n{n — i)(« — 2) 
1.2.3 



^«-3 + etc - • ( I0 ) 



By the aid of this equation we may now substitute for the several 
initial differences in equation (9) their values in terms of the successive 
values of the function. Also for any area divided into n sections by or- 
dinates, uniformly spaced a distance / apart, equation (9) will give the 
area in terms of /, ?i, and the several ordinates, when these latter are sub- 
stituted for the A's by means of eq. (10). 

Thus, for « = I, equation (9) becomes 



A =/(A + iA'A )= -fa + fii) (11) 



39 



6lO SURVEYING. 



which is the Trapezoidal Rule. 
For n = 2, 

A = I (2/fco + 2Z/'* + (f - I) J"* ) = /(#«+ Pi+ P 2 ) = - (*.+ 4^+ k 2 ), (12) 

which is called Simpson's §• _^#/<?. 

If / ; = 2/ = total length of figure, this formula becomes 

/' 
A = -(£<,+ 4^1 + ^2), • • (12a) 

which is the well-known form of the Pris?noidal Formula, and it would 
be that formula if areas were substituted for ordinates. 
If n 5= 3, 

>4 = f-^Ao + 3^i + 3^2 + *.), (13) 

which is called Simpson s f Rule. 
If » = 4, 

2/ 

^ = -- [7 (A* -f £ 4 ) + 32 {hi + £ 3 ) + 12^2] (14) 

45 

If n = 6, 

^ = / [6//0 + 18 A' h , + 2 7 J"/ lQ + M"'* + W^V/o + * §^0 + AW'^c]- 

If now the coefficient of ^ vi ^ be changed from -^ to T ^, which 
would not affect curves of a degree less than the sixth, the resulting 
equation, when the /i's are substituted for the A's, takes the following 
very simple form : 

A = -^[A, + A, + * 4 + A, + 5^ +*, + *.) + *,], . . (15) 

which is called Weddel's Rule. 

For a greater number of ordinates than seven, it is best to use either 
equation (12), (13), or (15) several times, as the formulae become very- 
complicated for n > 6. 



APPENDIX D. 



DERIVATION OF FORMULAE FOR COMPUTING GEOGRAPH- 
ICAL COORDINATES AND FOR THE PRO- 
JECTION OF MAPS.* 

Let Fig. 153 represent a distorted meridian section of the earth. 
Let a = the major and b the minor semi-axes. 

a — b 
Then e = = the ellipticity. 

The eccentricity is given by 

e 2 = 5—. whence 1 — <? 2 = — . 

a 1 a' 1 

The line nm = N is the normal to the curve at n ; 

the angle ncd — A. is the geocentric latitude ; 

while nld = L is the geodetic latitude. 

The geodetic latitude is always understood, as it is the latitude ob- 
tained from astronomical observations. 

It is desirable to find the length of the line nl, of the normal nm, and 
of the radius of curvature ft'r' , all in terms of e, L, and a. Also to find 
the geocentric latitude in terms of a, b, and L. 

To find nl, we have 



nl=V nd 2 + d?= |/^ + y~r 



For the ellipse, 



dy _ Px t 
dx a*y 



whence «/= |/y + ** = /■'' + «' -OW. 



* See Chapters XIV. and XV. for the use of the formulae. 



(2) 



6l2 



SURVEYING. 



But the equation of the ellipse in terms of its eccentricity is 



x 1 = a 2 



r 

i-e 1 



whence 



nl = VyV -f d> (i - e*)\ 
+ 




(3) 



Fig. 153. 

Squaring, remembering that y = nl sin L, we have, after reducing, 

a(i-* 2 ) 



nl = 



(I — e 1 sin 2 L)i 



(A) 



To find the length of the normal nm = N, we have 



ww : nl '.'. x : dl. 



But 



whence 



— dy P 

dl = nd tan dnl = y -— = — x = (1 — <r) x ; 



d!r a s 



nm = N — 



nl 



1 — £ 2 (1 — ^sin^Z)^' 



.... (4) 
. • • • (B) 



APPENDIX D. 613 



To "find A, the geocentric latitude in terms of a, b, and L, we have 
A. = tied \ Z = nld. 

Since both have the common ordinate nd, we may write 

tan A : tan Z :: dl : *£. 

But dl = —x from (4), and dc = .r, 

b* 
whence tan A. = -5 tan Z (C) 

To find the radius of curvature, R, we have, in general, 

^ = - — -— (5) 

dy 

"dx* 
Fortheelhpse, - = - -, -, and _ = __, 

whence ^t = in (A) 

To get this in terms of a, e, and L, we have, from Fig. 153, 

y 1 = nl sin 2 Z 



7i . . _ a 2 (1 - * 2 ) 2 sin 2 Z 



1 — <? 2 sin 2 Z 

Also from the equation of the ellipse in terms of its eccentricity we 
have 



Jz — sr2 



y _ a 2 (1 - sin 2 Z) 
1— <? 2 ~ 1 — ^ 2 sin 2 Z * 

We may now find 

a y _|_ ^ _ ** , 

^ ' 1 — <? 2 sin 2 Z 



614 



SURVEYING. 



or 



(«y + ^ 2 )l = 



« 3 3« 



(i - * 2 sin 2 Z)§' 



(7) 



Substituting this in (6), we obtain 



R = - a . 



>(!-«•) 



(i - * 2 sin 2 Z)§ (i — * 2 sin 2 Z)i* 



(D) 



The radius of curvature of the meridian, R, and the radius of curva- 
ture of the great circle perpendicular to a given meridian at the point 
where they intersect, which is the normal, N, are the most important 
functions in geodetic formulae. We will now derive the equations used 







Fig. 154. 



on the U. S. Coast and Geodetic Survey for computing geodetic positions 
from the results of a primary triangulation. 

In Fig. 154, let A and B be two points on the surface of the earth, 
which were used as adjacent triangulation-stations. The distance between 
them, the azimuth of the line AB at one of the stations, and the latitude 



APPENDIX D. 615 



and longitude of one station are supposed to be known ; the latitude and 
longitude of the other station, and the back azimuth of the line joining 
them, are to be found. 

Let L' = known latitude of B ; 

L — unknown latitude of A; 

K = known length of line AB reduced to sea-level; 

s = length of arc AB = — -; 

Z' = known azimuth of BA at B; 
Z= unknown azimuth of AB at A\ 
M' — known longitude of B\ 
M = unknown longitude of A. 

The angle APB formed by the two meridional planes through A and 
B is the difference of longitude in M— M' — AM. 

The difference of latitude is, L — L' = AL = Bl in the figure. Al is 
the trace of a parallel of latitude through A and / is its intersection with 
the meridian through B. AP' is the trace of a great circle through A 
perpendicular to the meridian through B, and P' is the point of its inter- 
section with that meridian. 

The normals are B n '= N' and A n = N. The radii of curvature are 
■Br '= R' and A„ = R. 

The latitude and longitude of A, and the azimuth of the line AB from 
A towards B, can now be found by solving the spherical triangle APB. 
Thus L = 90 - AP; M — M' - Al; and Z = 180 - PAB. 

Although the line AB lies on the surface of a spheroid, if a sphere be 
conceived such that its surface is tangent internally to the surface of the 
spheroid on the parallel of latitude passing through the middle point of 
the line AB, then this line will lie so nearly in the surface of the sphere, 
that no appreciable error is made by assuming it to be in its surface. The 
triangle ABP then becomes a triangle on the surface of the tangent 
sphere, and hence is a true spherical triangle. The sphere is defined by 
taking its radius equal to the normal to the meridian at the mean lati- 
tude of the points A and B. Since this mean latitude is unknown, the 
formulae are first derived for the latitude of B, L', and then a correction 
applied to reduce it to the mean latitude. 



THE DIFFERENCE OF LATITUDE. 

Let it first be required to find L from L', or find AL =L — L'. 
If we write /, /', for the co-latitudes of L, L', and 2' for 180 — Z\ we 
have, from the spherical triangle ABP, 

cos / = cos /' cos s -f- sin /' sin s cos z' (8) 



6i6 



SURVEYING. 



By means of Taylor's Formula we may find the value of / in ascending 
powers of s, and since s is always a very small arc in terms of the radius, 
usually from 20 to 60 minutes, the series will be rapidly converging 

By means of Taylor's Formula, we may at once write 



/ r _l dl ' 



1 d-i 



+ - - — s l 4- - 



2 ds 



6 ds* 



s z -j- etc. 



(9) 



We will use but the first three terms of this development, the fourth 
term being used only in the largest primary triangles. 

The derivation of the successive differential coefficients of / with 
respect to s is the most difficult portion of this general development. If 
s be supposed to vary, then /and z both must vary, and they are all im- 
plicit functions of each other. These coefficients are therefore best 

/ found geometrically, as follows : In Fig. 155, 

Let AB = BC = ds = differential portions of the line 
AB — s in Fig. 154; 

AD = — d'l = change in AP(= I) due to first 
change -\- ds in s; 

DE = — d"l — change in AP due to second 
change -f- ds in s. 




Then 



CP = EP = / - d'l - d"l. 



Let angles PAB — z and PBC = z. 

Since the line s is a straight line upon the 
surface of the sphere (lies in the plane of a great 
circle), the angle z' is greater than the angle z 
by the amount of convergence of the meridians 
AP and BP at that point. This convergence 
is the angle their tangents AP' and BP' make 
at their intersection, or z' = z + AP'B. 

The lines BD and CE are parallels of lati- 
tude through the points B and C. They cut all 
meridians at right angles. 
We then have, from the geometrical relations, 



Fig. 155. 



AD = AB cos z, and DE = BC cos z , 

or d'l = — ds cos z, and d"l — — ds cos z' (10) 

Since z' is greater than z, d'l is greater than d"l. But the difference 



APPENDIX D. 6l7 



between these successive first differentials is the value of the correspond- 
ing second differential ; hence we may write 

d*l = d"l - d'l. 

We will now find the value of d"*l. First, we have 

d" I = — ds cos 2' = — ds cos (z -\- AP' B). 

The angle AP'B is equal to the arc DB, whose centre is N t reduced 
to the new radius BP' t or 

R P 
angle AP'B = ds sin z — — = ds sin z cot /, 

BJy 

since the angle BNP is the co-latitude, /. 
We have then 

d"l = — ds cos (z -{- ds sin z cot /) 

— — ds [cos z cos (ds sin 2 cot /) — sin z sin (ds sin 2 cot /)] 
= — ds [cos 2 — sin 2 z cot /<&]. 

Since 

cos (ds sin 2 cot /) = I, and sin (ds sin z cot /) = ^j sin 2 cot / ; 

. • . d"l = — cos zds -j- sin 2 2 cot Ids' 2 , 

and</*/= </"/- </7= + sin 2 2 cot /</r 2 (11) 

From equations (10) and (11) we may at once write 

— = — cos 2, and — - = -f sin 2 2 cot /. . . . . (12) 
ds ds* 

Substituting these values in (9), we have 

/ — /' = — s cos 2 -|- \s* sin 2 2 cot / -f- etc. 

Now, replacing /, /', and z, by L, L', and Z, we have 

U — L — s cos Z-f-ir 2 sin 2 Ztan Z. . . . . . (T3) 

Here s is expressed in arc to a radius of unity. 



6l8 SURVEYING. 



Referring it now to the radius N, we have .y = — , where K is the length 

of the arc s in any unit, N being the length of the normal nm in Fig. 153, 
given in the same unit. 

Substituting these in (13), we have 

,, , ^"cos Z . 1 R~i sins Ztan L , „ 

z - z = -^r-+2 — Jn — (I4) 

This gives the difference of latitude in units of arc in terms of radius N. 
But differences of latitude are measured on a sphere whose radius is 
the radius of curvature of the meridian at the middle latitude. Since we 
do not yet know the middle latitude, we can use the known latitude U 

and afterwards correct to . 

2 

Changing to a sphere whose radius is R, and dividing by the arc of 
1" in order to get the result in seconds, we have 

L' - L = — 8L = „ K -r, cos Z 4- - -— — sin 2 Z tan Z. (15) 

R arc 1" ~ 2 RN arc 1" v 3/ 

_, _ 1 _ tan Z 

If we let B = -=■ rr» and C = 



Rarci" 2RJVarci'" 

we may write — SL = ^ cos Z-B -f- R~* sin 2 Z-C. (16) 

To reduce this to what it would be if the mean latitude had been used 
we have to correct it for the difference in the radii of curvature. R& and 
R m , at the latitude L and the middle latitude respectively. If AL be the 
true difference of latitude when R m is used, and SL be the difference 
when R L is used, we would have 

AL:SL::R L : R m , 
AL = SL^- = SL (j + **=-*!*) = SL (1 +^). 

Km ' Km ' * Km ' 

dRm 

To reduce SL to AL, therefore, we must add the quantity SL 
Now R = 



R, 
a{\ — e*) 



(I — e* sin 2 Z)f 

,_ a (1 — ei) (3<f 8 sin Z cos L) rT 

whence dR = — — - o . „ T< k 'dL. 

(I — e 2 sin a Z)l 



APPENDIX D. 619 



Here dL is the difference in latitude between one extremity of the 
line s and its middle point, or dL = %dL, as given in eq. (16), hence 

8L M?\ = $e*smLcosL 

V R I 1 — ei sin 2 L v '■ 

_ \ei sin L cos L sin 1" 

If we put D = g . „ , 

r 1 — * 2 sin 2 Z 

the corrective term becomes 

Km 

whence we finally obtain 

- AL = K cos Z-B + K* sin 2 Z.C+(SZ) 2 . D, . . . (D) 

where SL is given by (16), or it is the value of the first two terms in the 
right member of this final equation. For distances less than 12 miles 
the first term only may be used as giving the value of dL. 

The values of the constants B, C, and D are given for every minute 
of latitude from 23 to 65 in Appendix No. 7 of the U. S. Coast and 
Geodetic Report for 1884. This Appendix can be obtained by applying 
to the Superintendent. 

For distances of 12 miles or less, using the first term only for SL, 
equation (18) becomes 

AL = K cos Z(B + K cos Z-D) + K* sin 2 Z- C. . . . (E') 
THE DIFFERENCE OF LONGITUDE. 

In the triangle APB, Fig. 154, the three sides and the angle at the 
known station B are known. To find AM = angle APB, we have, 
therefore, 

sin PA : sin AB :: sin PBA : sin APB, 
or sin / : sin s : : sin z : sin AM. 



* In the U. S. Coast and Geodetic Survey Report for 1884, Appendix 7, p. 
326, this term is given with its denominator raised to the f power, and the 
tabular values of D are computed accordingly. The development there given 
is laborious and approximate, but the error is not more than 0.001 of the value 
of this term, which is itself very small. 



620 



SURVEYING. 



But s = -*f where N is the normal An, Fig. 154; and if we assume 



that the arc s is proportional to its sine, we have 



AM 



J? sin Z 



A r cos Zarc 1" 



(18) 



where AM is expressed in seconds of arc. 

1 



If we put 
this equation becomes 



AM = 



N arc 1"' 

Z^sin Z . A 
cos Z 



(F) 



In order to correct for the assumption that the arc is proportional to 
its sine, a table of the differences of the logarithms of arcs and sines is 
given in the U. S. C. and G. Report for 1884, p. 373, with instructions 
for its use on p. 327. 



THE DIFFERENCE OF AZIMUTH. 



In the spherical triangle APB, Fig. 154, we have, from spherical 
trigonometry, 



cot \{PBA + PAB)* = tan \BPA 



or 



cot *(*' + *) = tan \ (- AM\) 



cos j(BP -f AP) 
cos }(BP~-APy 

cos-H/'-l-/) 



cos^(/'— /) 

, A „ sin \(L + L) 

= - tan \AM jry—-/, 

cos i{L — Z) 

But z = 180 — Z, 

therefore cot £(i8o° - Z -\- z') ~ tan \{Z - z') = tan UAZ), 

sin i(Z' + Z) 



whence 



— tan \AZ — \2,n\AM- 



sin £(Z' - L) ' 



(19) 



*Chauvenet's Spherical Trigonometry, eq. (127). 

f Increments of M are measured positively towards the west. 



APPENDIX D. 621 



It will be seen that since the azimuth Z of a line is measured from the 
south point in the direction S.W.N.E., the azimuth of the line BA 
from B towards A (forward azimuth) is the angle PBA + 180 = Z\ 
while the azimuth of the same line from A is 180 — PAB = Z. Also, 
that AZ = Z + 180 - Z'. 

Assuming that the tangents \AZ and \AM are proportional to their 
arcs, and putting L m for the middle latitude, we have 

a r, a sin JLtn ,_. 

- AZ= Am — j— (G) 

cos \AL v ' 

The U. S. Coast Survey Tables are based on the following semi- 
diameters : 

a = 6 378 206 metres, 
b = 6 356 584 " 

or a : b :: 294.98 : 293.98. 

See Appendix No. 7, U. S. Coast and Geodetic Survey, for tabular 
values of constants and forms for reduction. 



TABLES. 



TABLE I. 

Trigonometric Formula. 



Trigonometric Functions. 

Let A (Fig. 107) = angle BAG = arc BF, and let the radius AF — AB == 
AH= 1. 



"We then have 




sin A 


= BG 


cos A 


= AG 


tan A 


= DF 


cot A 


= HG 


sec A 


= AD 


cosec A 


= AG 


versin A 


= CF- BE 


covers A 


= BK=HL 


exsec A 


= BD 


coexsec A 


= BG 


chord A 


= BF 


chord 2 A 


= BI= 2BC 




Fig. 107. 



In the right-angled triangle ABC (Fig. 107) 
Let AB = c t AG = 6, and .BC = a. 
We then have : 



1. sin A 

2. cos A 

3. tan .4 

4. cot ^4 

5. sec A 



— = cos B 
c 



= sin B 



= — = cot i? 



= — = tan B 
a 



— cosec B 



6. cosec A = — = sec B 

a 

7. vers 4 = = covers B 



i. exsec A = — , — = coexsec B 
o 



9. covers A 



= versin B 



10. coexsec A = = exsec jB 



11. a = c sin yl = b tan ^4 

12. b = c cos-4 = a cot ^4. 
iq „ _ a b 

16. C = — 



21. area = 



sin -4 cos .4 

14. a = c cos B = b cot 5 

15. b = c sin B = a tan 5 

1G. c = 5 = . — = 

cos B sm i? 

17 - a = V (c + 6) (c - 6) 

18. 6 = ^ (c + a) (c - a) 

19. c s= |/ a aqT52 

20. (7 = 90° = .4 + 5 

2 



626 



SURVEYING. 



TABLE \.— Continued. 
Trigonometric Formulae. 







Solution of Oblique Triangles. 

a/ * No 


* 




Fig. 108. 


22 


GIVEN. 


SOUGHT. 


FORMULAE. 


A, B,a 


C, 6, c 


r> 1 ono ( A , | T>\ h c.| n p 


7 ' sin A 










sin A 


23 


A, a, b 


B,C, c 


sin B = — n — .6, C = 180° -{A A- B\ 
a 

a sinC 


sin -4 ' ' 


24 


C,a,b 


VzU + B) 


J4 (4 + B) = 90° - y 2 


25 




}4(A-B) 


tany 2 (A-B)^ ( ^~ > t&ny 2 U + B) 


26 




A,B 


A = )6 (A + B) + H (A - B), 

B = y z u + B)-y 2 u-B) 


27 




c 


1 cos^(^--6) sin y 2 (A — B) 


28 
29 

30 
31 


a, b f c 


area 

A 


K - y 2 a b sin C. 


Let * = y % (a + b -f c) ; sin % A = j/- — ^ ^ - C) 


,, . /s(s — a) , .. . /(s-b)(s-c) 
co S ^=|/ bc ;tan^^=|/- s - (s _ a) 


. , 2V s (s - a)(s-b)(s-c) 


EmA ~ be 


32 




area 


A 2 (s -b)(s- c) 

vers A — ~ : 

bc 


K = Vs (s - a) (s - b) (s - c) 


33 


A, B, C, a 


area 


a 2 sin 2? . sin C 

iv — — . . 
2 sin A 



TABLES. 



627 



TABLE I — Continued. 
Trigonometric Formulae. 



GENERAL FORMULAE. 



34 

35 
36 

37 

38 
39 

40 
41 

42 

43 
44 

45 

46 

47 
48 

49 

50 

51 
52 



sin A = 

sin A = 

sin A = 

cos A — 

cos A — 

cos A = 

tan A = 

tan A = 

tan A = 

cot A as 
cot A = 



1 



= VI — cos 2 A 



tan .4 cos A 



cosec J. 

2 sin y% A cos ^ A = vers -4 cot J^ 41 



4/ ^ vers 2 .4 = 4/^ (1 — cos 2 ^4) 

1 



= V 1 — sin 2 J. = cot .4 sin ^4 



sec 4. 

1 — vers ^4 = 2 cos 2 y»A—\ = 1 — 2 sin 9 \& -4 



cos 2 y 2 A — sin 2 ^ A = V% + %cos2A 
1 sin .4 



cot .4 



cos -4 



= V sec 2 .4 — 1 



S 



cos 2 .4 
1 — cos 2 >4 



4/ 1 — cos 2 .4 
cos ^4 



sin 2 A 
1 + cos 2 A 



vers 2 4. 



sin 2 .4 sin 2 ^4 

1 cos A 



= exsec -4. cot \& A 



tan -4 sin A 

sin 2 .4 sin 2^4 

1 — cos 2 .4 



= V cosec 2 A — 1 

1 + cos 2 A 



vers 2 .4 



sin 2 A 



cot .4 = — 



vers A 
vers -4 
exsec A 



tan ^2^ 
exsec ^4 

1 — cos .4 = sin A tan 14 A = 2sin a J£.4 

exsec A cos .4 



^ h * 4 *- 1 y a vers-4 

= sec .4 — 1 = tan A tan ^ 4. = , 



/l — cos .4 / 

= f — ^— = y- 



sin J^u4 

sin 2 -4 = 2 sin 4. cos A 



vers .4 



/ 



1 + cos_4L 
2 



cos ^j.4 

cos 2 ^4 = 2 cos* ^4 — 1 = cos 2 .4 — sin 2 ,4 = 3 — 2 sin* A 



628 



SUR VE YING. 



TABLE I. — Continued. 
Trigonometric Formula. 



General Formula. 



53. tan %& A = ;— j- j = cosec 

'* 1 4- sec A 



. . 1 — cos -4 . /l — cos.4 

4 — COt A = - • = 4/ r— j ~. 

smA V 1 4- cob -4 



54. tan 2 A = 



55. cot. J^j.4 



56. cot 2 .4 = 



vers*&A 



2 tan .4 
1 - tan 3 ^ 



sin ,4 _ 1 + cos A 
vers .4 " sin A 



cot* A — l 



cosec A — cot A 



57, 

58. 

59. exsec l£A = 



2 cot A 
%4 vers .4 



1 — cos A 



1+^1 — H vers A 2 4- V2 (1 + cos .4) 
vers 2 A = 2 sin a ,4 

1 — cos A 



60. exsec 2 .4 



(1 4- cos A) + V2 (1 4- cos A) 
tan 2 4 



1 — tan 2 A 
sin (A ± B) = sin A . cos 2? ± sin B . cos .4 
cos {A ± B) = cos -4 . cos i? T sin .4 . sin B 
sin .4 4- sin B = 2 sin ^ (A + B) cos ^ (A — B) 
sin ^4 — sin B = 2 cos }/% (A 4- JB) sin }4(A — B) 
cos ^4 4- cos B = 2 cos \i (A 4- £) cos )4(A- E) 
cos £ — cos A = 2 sin J^ (A + 5) sin 14 (A — B) 

sin 9 ^4 — sin 2 5 = cos 2 B — cos 2 A = sin (4 4- B) sin (-4 — B; 
cos 3 A — sin 2 5 = cos {A + B) cos {A — B) 



. At*. d sin (.4 4- B) 

tan -4 4- tan B = ■ ^ — ' — ~ 

cos A . cos 5 



tan.4-tanl? = siD ( / ~ % 

cos A , cos B 



TABLES, 



629 



TABLE II. 
For Converting Metres, Feet, and Chains. 



Metres to Feet. 


Feet 


to Metres and 


Chains. 


Chains 


to Feet. 


Metres. 


Feet. 


Feet. 


Metres. 


Chains. 


Chains. 


Feet. 


I 


3.28087 


I 


O.304797 


O.OI51 


O.Ol 


0.66 


2 


6.56174 


2 


O.609595 


.0303 


.02 


1.32 


3 


9.84261 


3 


O.914392 


•0455 


•03 


I.98 


4 


13.12348 


4 


1.219189 


.0606 


.04 


2.64 


5 


16.40435 


5 


I.523986 


.0758 


.05 


3-30 


6 


19.68522 


6 


I.828784 


.0909 


.06 


3-96 


7 


22 . 96609 


7 


2.I3358I 


.1061 


.07 


4.62 


8 


26 . 24695 


8 


2.438378 


.1212 


.08 


5-28 


9 


29.52782 


9 


2-743175 


.I364 


.09 


5-94 


10 


32.80869 


10 


3.047973 


•1515 


.10 


6.60 


20 


65.61739 


20 


6.095946 


.3030 


.20 


13.20 


30 


98.42609 


30 


9.143918 


•4545 


• 30 


19.80 


40 


131.2348 


40 


12.19189 


.6061 


.40 


26.40 


50 


164.0435 


50 


15.23986 


•7576 


• 50 


33-00 


60 


I96.8522 


60 


18.28784 


.9091 


.60 


39.60 


70 


229.6609 


70 


21.33581 


1 . 0606 


• 70 


46.20 


80 


262.4695 


80 


24.38378 


1.2121 


.80 


52.80 


go 


295.2782 


90 


27-43I75 


- 1.3636 


.90 


59-40 


100 


328.0869 


100 


30.47973 


I.5I5I 


I 


66.00 


200 


656.1739 


100 


60.95946 


3.0303 


2 


132 


300 


984.2609 


300 


91.43918 


4-5455 


3 


198 


400 


1312.348 


400 


121. 9189 


6 . 0606 


4 


264 


500 


1640.435 


500 


I52.3986 


7-5756 


5 


330 


600 


I968.522 


600 


182.8784 


9.0909 


6 


396 


700 


2296.609 


700 


2I3-358I 


10.606 


7 


462 


800 


2624.695 


800 


243-8378 


12. 121 


8 


528 


900 


2952.782 


900 


274-3I75 


13.636 


9 


594 


1000 


3280.869 


1000 


304-7973 


I5-I5I 


10 


660 


2000 


6561.739 


2000 


609.5946 


30.303 


20 


1320 


3000 


9842.609 


3000 


914.3918 


45-455 


30 


1980 


4000 


13123.48 


4000 


I2I9.189 


60.606 


40 


2640 


5000 


16404.35 


5000 


1523.986 


75-756 


50 


3300 


6000 


19685.22 


6000 


1828.784 


90.909 


60 


3960 


7000 


22966.O9 


7000 


2I33-58I 


106.06 


70 


4620 


8000 


26246.95 


8000 


2438.378 


121. 21 


80 


5280 


9000 


29527.82 


9000 


2743.175 


136.36 


90 


5940 



630 



SUR VE YING. 



TABLE III. 

Logarithms of Numbers. 



(A 







1 


2 


3 


4 


5 


6 


*7 


8 


9 


Proportional Parts. 












































1 
4 


2 

8 


3 

12 


4 

17 


5 

21 


6 

25 


7 
29 


8 

33 


9 

37 


10 


.0000 


.0043 


.0086 


.0128 


.0170 


.0212 


.0253 


.0294 


•0334 


•0374 


11 


.0414 


•0453 


.0492 


•053 1 


.0569 


.0607 
.0969 


.0645 


.0682 


.0719 


•0755 


4 


8 


11 


15 


19 


2 3 


26 


SO 


34 


12 


.0792 


.0828 


.0864 


.0899 


•o934 


.1004 

•1335 


.1038 


.1072 


.1106 


3 


7 


10 


14 


17 


21 


24 


28 


3 1 


13 


•"39 


•"73 


.1206 


.1239 


.1271 


•^QS 


• 1367 


>*$<& 


.1430 


3 


6 


10 


*3 


16 


19 


23 


26 


29 


14 


.1461 


.1492 


•1523 


•1553 


.1584 


.1614 


.1644 


•1673 


• I 7°3 


.1732 


3 


6 


9 


12 


IS 


18 


21 


24 


27 


.J 


.1761 


.1790 


.1818 


.1847 


•1875 


.1903 


•1931 


•1959 


.1987 


.2014 


3 


6 


8 


11 


14 


t-7 


20 


22 


25 


16 


.2041 


.2068 


• 2095 


.2122 


.2148 


•2175 


.2201 


.2227 


.2253 


.2279 


3 


5 


8 


11 


13 


16 


18 


21 


24 


T-l 


.2304 


.2330 


•2355 


.2380 


.2405 


.2430 


•2455 


.2480 


.2504 


.2529 


2 


5 


7 


10 


12 


15 


17 


20 


22 


18 


•2553 


•2577 


.2601 


.2625 


.2648 


.2672 


.2695 


.2718 


.2742 


.2765 


2 


5 


7 


9 


12 


14 


16 


19 


21 


ig 


.2788 


.2810 


.2833 


.2856 


.2878 


.2900 


.2923 


•2945 


.2967 


.2989 


2 


4 


7 


9 


ii 


13 


16 


18 


20 


20 


.3010 
.3222 


• 3°3 2 


•3°54 


•3075 


.3096 


• 3"8 


•3 I 39 


.3160 


.3181 


.3201 


2 


4 


6 


8 


11 


13 


15 


17 


19 


21 


•3243 


.3263 


.3284 


•3304 


•3324 


•3345 


•3365 


•3385 


•3404 


2 


4 


6 


8 


to 


12 


14 


16 


18 


22 


•3424 


•3444 


•3464 


.3483 


.3502 


•3522 


•354 1 


•3560 


•3579 


•3598 


2 


4 


6 


8 


10 


12 


J 4 


15 


*7 


23 


•3 6r 7 


•3636 


.3655 


•3674 


.3692 


•37" 


•3729 


•3747 


.3766 


•3784 


2 


4 


6 


7 


9 


11 


13 


IS 


17 


24 


.3802 


.3820 


•3838 


.3856 


•3874 


.3892 


•3909 


•3927 


•3945 


.3962 


2 


4 


5 


7 


9 


11 


12 


14 


16 


25 


•3979 


•3997 


.4014 


.4031 


.4048 


.4065 


.4082 


.4099 


.4116 


•4i33 


2 


3 


5 


7 


9 


10 


12 


14 


15 


26 


.4150 
•43*4 


.4166 


.4183 


.4200 


.4216 


.4232 


•4249 


.4265 


.4281 


.4298 


2 


3 


5 


7 


8 


10 


" 


13 


15 


27 


•4330 


.4346 


.4362 


•4378 


•4393 


.4409 


•4425 


.4440 


•4456 


2 


3 


5 


6 


8 


9 


" 


13 


14 


23 


•4472 


•4487 


.4502 


.4518 


•4533 


•4548 


•45 6 4 


•4579 


•4594 


.4609 


2 


3 


5 


6 


8 


9 


" 


12 


14 


29 


.4624 


•4 6 39 


•4654 


.4669 


.4683 


.4698 


•4713 


.4728 


.4742 


•4757 


1 


3 


4 


6 


7 


9 


10 


12 


*3 


30 


•4771 


.4786 


.4800 


.4814 


.4829 


•4843 


•4857 


.4871 


.4886 


.4000 




3 


4 


6 


7 


9 


10 


11 


13 


31 


.4914 


.4928 


.4942 


•4955 


.4969 


•4983 


•4997 


.5011 


.5024 


•5038 




3 


4 


6 


7 


8 


10 


11 


12 


32 


•5051 
•5185 


•5065 


•5079 


.5092 


•5105 


•5"9 


■5132 


•5145 


•5159 


.5172 




3 


4 


5 


7 


8 


9 


11 


12 


33 


.5198 


.5211 


.5224 


•5237 


•5250 


•5263 


•5276 


.5289 


.5302 




3 


4 


5 


6 


8 


9 


10 


12 


34 


•53i5 


.5328 


•534° 


•5353 


.5366 


•5378 


•5391 


•5403 


.5416 


.5428 




3 


4 


5 


6 


8 


9 


10 


11 


35 


•5441 


•5453 


•5465 


•5478 


•5490 


•5502 


•5514 


•5527 


•5539 


•555i 




2 


4 


5 


6 


7 


9 


10 


11 


36 


•5563 


•5575 


•5587 


•5599 


.5611 


.5623 
•574o 


•5635 


•5647 


.5658 


.5670 




2 


4 


5 


6 


7 


8 


10 


11 


37 


.5682 


•5694 


•5705 


•5717 


•5729 


•5752 


•5763 


•5775 


•5786 




2 


3 


5 


6 


7 


8 


9 


10 


38 


•5798 


.5809 


.5821 


•5832 


•5843 


•5£55 


.5866 


•5877 


.5888 


•5899 




2 


3 


5 


6 


7 


8 


9 


10 


39 


•59" 


.5922 


•5933 


•5944 


•5955 


.5966 


•5977 


.5988 


•5999 


.6010 




2 


3 


4 


5 


7 


8 


9 


10 


40 


.6021 


.6031 


.6042 


•6053 


.6064 


.6075 


.6085 


.6096 


.6107 


.6117 




2 


3 


4 


5 


6 


8 


9 


10 


4i 


.6128 


.6138 


.6149 


.6160 


.6170 


.6180 


.6191 


.6201 


.6212 


.6222 




2 


3 


4 


5 


6 


7 


8 


9 


42 


.6232 


.6243 


• 6 253 


.6263 


.6274 


.6284 


.6294 


.6304 


.6314 


•6325 




2 


3 


4 


5 


6 


7 


8 


9 


43 


•6335 


•6345 


• 6 355 


•6365 


.6375 


.6385 


•6395 


.6405 


.6415 


.6425 




2 


3 


4 


5 


6 


7 


8 


9 


44 


•6435 


.6444 


•6454 


.6464 


.6474 


.6484 


•6493 


.6503 


•6513 


.6522 




2 


3 


4 


5 


6 


7 


8 


9 


45 


•6532 


.6542 


•6551 


.6561 


•6571 


.6580 


.6590 


•6599 


.6609 


.6618 




2 


3 


4 


5 


6 


7 


8 


9 


46 


.6628 


.6637 


.6646 


.6656 


.6665 


.6675 


.6684 


.6693 


.6702 


.6712 




2 


3 


4 


5 


6 


7 


7 


8 


47 


.6721 


.6730 


•6739 


.6749 


.6758 


.6767 


.6776 


•6785 


.6794 


.6803 




2 


3 


4 


5 


5 


6 


7 


8 


48 


.6812 


.6821 


.6830 


.6839 


.6848 


.6857 


.6866 


.6875 


.6884 


.6893 




2 


3 


4 


4 


5 


6 


7 


8 


49 


.6902 


.6911 


.6920 


.6928 


• 6 937 


.6946 


•6955 


.6964 


.6972 


.6981 




2 


3 


4 


4 


5 


6 


7 


8 


50 


.6990 


.6998 


.7007 


.7016 


.7024 


•7033 


.7042 


.7050 


•7°59 


.7067 




2 


3 


3 


4 


5 


6 


7 


8 


5i 


.7076 


.7084 


.7093 


.7101 


.7110 


.7118 


.7126 


•7135 


•7i43 


•7 X 52 




2 


3 


3 


4 


S 


6 


7 


8 


52 


.7160 


.7168 


.7177 


•7185 


•7193 


.7202 


.7210 


.7218 


.7226 


•7235 




2 


2 


3 


4 


5 


6 


7 


7 


53 


.7243 


•7251 


•7259 


.7267 


•7275 


.7284 


.7292 


.7300 


.7308 


•73i6 




2 


2 


3 


4 


5 


6 


6 


7 


54 


•73 2 4 


•7332 


•734° 


.7348 


•735^ 


•7364 


•7372 


.7380 


•7388 


•7396 




2 


2 


3 


4 


5 


6 


6 


7 



TABLES. 



63I 



TABLE III.— Continued. 
Logarithms of Numbers. 



O 





1 


2 


3 


4 


5 


6 


7 


8 


9 


Proportional 


Parts. 












5 






: 


rt 






















1 


2 


3 


4 


6 


7 


89 


55 


.7404 


.7412 


.7419 


•7427 


•7435 


•7443 


•745i 


•7459 


.7466 


•7474 


2 


2 


3 


4 


5 


5 


1 
6 7 


56 


.7482 


.7490 


•7497 


•7S05 


■75i3 


•752o 


.7528 


•7536 


•7543 


•755i 




2 


2 


3 


4 


5 


5 


6 7 


57 


•7559 


.7566 


•7574 


• 7582 


•7589 


•7597 


.7604 


.7612 


.7619 


.7627 




2 


2 


3 


4 


5 


5 


6 


7 


58 


•7634 


.7642 


.7649 


•7657 


.7664 


.7672 


.7679 


.7686 


.7694 


.7701 




1 


2 


3 


4 


4 


5 


6 


7 


59 


.7709 


.7716 


•7723 


•773 1 


•7738 


•7745 


•7752 


.7760 


.7767 


•7774 




1 


2 


3 


4 


4 


5 


6 


7 


60 


.7782 


.7789 


.7796 


•7803 


.7810 


.7818 
.7889 

•7959 
.8028 
.8096 


.7825 


.7832 


•7839 


.7846 




1 


2 


3 


4 


4 


5 


6 


6 


61 


.7853 


.786c 


.7868 


•7875 


.7882 


.7896 


•79°3 


.7910 


.7917 




1 


2 


_ 


4 


4 


5 


6 


6 


62 


.7924 


•793 1 


•7938 


•7945 


•7952 


.7966 


•7973 


.7980 


.7987 




1 


2 


3 


3 


4 


5 


6 


6 


63 


•7993 


.8000 


.8007 


.8014 


.8021 


•8035 


.8041 


.8048 


•8055 




1 


2 


3 


3 


4 


5 


5 


6 


64 


.8062 


.8069 


.8075 


.8082 


.8089 


.8102 


.8109 


.8116 


.8122 




1 


2 


3 


3 


4 


5 


5 


6 


65 


.8129 


.8136 


.8142 


.8149 


.8156 


.8162 


.8169 


.8176 


.8182 


.8189 




1 


2 


3 


3 


4 


5 


5 


6 


66 


.8195 


.8202 


.8209 


.8215 


.8222 


.8228 


•8235 


.8241 


.8248 


.8254 




1 


2 


3 


3 


4 


5 


5 


6 


67 


.8261 


.8267 


•8274 


.8280 


.8287 


.8293 


.8299 


.8306 


.8312 


.8319 




1 


2 


3 


3 


4 


5 


5 


6 


63 


•8325 


•8331 


•8338 


•8344 


•8351 


•8357 


•8363 


.8370 


.8376 


.8382 




1 


2 


3 


3 


4 


4 


5 


6 


69 


.8388 


•8395 


.8401 


.8407 


.8414 


.8420 


.8426 


.8432 


•8439 


•8445 




1 


2 


2 


3 


4 


4 


5 


6 


70 


.8451 


•8457 


.8463 


.8470 


.8476 


.8482 


.8488 


.8494 


.8500 


.8506 




1 


2 


2 


3 


4 


4 


5 


6 


71 


•8513 


.8519 


•8525 


•8531 


•8537 


.8543 


•8549 


•8555 


.8561 


•8567 




1 


2 


2 


3 


4 


4 


5 


5 


72 


•8573 


••8579 


•8585 


.8591 


•8597 


.8603 


.8609 


.8615 


.8621 


.8627 




1 


2 


2 


3 


4 


4 


5 


5 


73 


•8633 


.8639 


.8645 


.8651 


.8657 


.8663 


.8669 


•8675 


.8681 


.8686 




1 


2 


2 


3 


4 


4 5 


5 


74 


.8692 


.8698 


.8704 


.8710 


.8716 


.8722 


.8727 


•8733 


•8739 


•8745 




1 


2 


2 


3 


4 


4 


5 


5 


75 


•8751 


.8756 


.8762 


.8768 


.8774 


.8779 


.8785 


•8791 


.8797 


.8802 




1 


2 


2 


3 


3 


4 


5 


5 


76 


.8808 


.8814 


.8820 


.8825 


.8831 


.8837 


.8842 


.8848 


.8854 


.8859 




1 


2 


2 


3 


3 


4 


5 


5 


77 


.8865 


.8871 


.8876 


.8882 


.8887 


.8893 


.8899 


.8904 


.8910 


.8915 




1 


2 


2 


3 


3 


4 


4 


5 


78 


.8921 


.8927 


.8932 


.8938 


•8943 


•8949 


•8954 


.8960 


.8965 


.8971 




1 


2 


2 


3 


3 


4 


4 


5 


79 


.8976 


.8982 


.8987 


•8993 


.8998 


.9004 


.9009 


.9015 


.9020 


.9025 




1 


2 


2 


3 


3 


4 


4 


5 


80 


.9031 


.9036 


.9042 


.9047 


•9053 


.9058 


.9063 


.9069 


.9074 


.9079 




1 


2 


2 


3 


3 


4 


4 


5 


81 


.9085 


.9090 


.9096 


.9101 


.9106 


.9112 


.9117 


.9122 


.9128 


•9 1 33 




1 


2 


2 


3 


3 


4 


4 


5 


82 


.9138 


•9*43 


.9149 


•9154 


•9^59 


.9165 


.9170 


•9 J 75 


.9180 


.9186 




1 


2 


2 


3 


3 


4 


4 


5 


83 


.9191 


.9196 


.9201 


.9206 


.9212 


.9217 


.9222 


.9227 


.9232 


.9238 




1 


2 


2 


3 


3 


4 


4 


5 


84 


•9 2 43 


.9248 


•9253 


■9258 


.9263 


.9269 


.9274 


•9279 


.9284 


.9289 




1 


2 


2 


3 


3 


4 


4 


5 


85 


.9294 


•9299 


•9304 


•93°9 


•9315 


• 9320 


•9325 


■9330 


•9335 


•9340 


1 


1 


2 


2 


3 


3 


4 


4 


5 


86 


•9345 


•9350 


•9355 


.9360 


•9365 


•9370 
.9420 


•9375 


.9380 


•9385 


•939° 




1 


2 


2 


3 


3 


4 


4 


5 


87 


•9395 


.9400 


•9405 


.9410 


•94i5 


•9425 


•943° 


•9435 


.9440 





1 


1 


2 


2 


3 


3 


4 


4 


88 


•9445 


•945o 


•9455 


.9460 


■9465 


.9469 


•9474 


•9479 


.9484 


•9489 





1 


1 


2 


2 


3 


3 


4 


4 


89 


•9494 


•9499 


.9504 


•9509 


•95i3 


.9518 


•9523 


.9528 


•9533 


•9538 





1 


1 


2 


2 


3 


3 


4 


4 


90 


•9542 


•9547 


•9552 


•9557 


.9562 


.9566 


•957i 


•9576 


.9581 


.9586 





1 


1 


2 


2 


3 


3 


4 4 


91 


•959o 


•9595 


.9600 


.9605 


.9609 


.9614 


.9619 


.9624 


.9628 


•9 6 33 





1 


1 


2 


2 


3 


3 


4 4 


92 


.9638 


•9 6 43 


.9647 


.9652 


•9657 


.9661 


.9666 


.9671 


•9675 


.9680 





1 


1 


2 


2 


3 


3 


4 4 


93 


.9685 


.9689 


.9694 


.9699 


•9703 


.9708 


•9713 


.9717 


.9722 


.9727 





1 


1 


2 


2 


3 


3 


4 


4 


94 


•9731 


•973 6 


.9741 


•9745 


•975o 


•9754 


•9759 


•9763 


.9768 


•9773 





1 


1 


2 


2 


3 


3 


4 


4 


95 


•9777 


.9782 


.9786 


.9791 


•9795 


.9800 


.9805 


.9809 


.9814 


.9818 





1 


1 


2 


2 


3 


3 


4 


4 


96 


.9823 


.9827 


•9832 


.9836 


.9841 


■9845 


.9850 


•9854 


•9859 


.9863 





1 


1 


2 


2 


3 


3 


4 


4 


97 


.9868 


.9872 


.9877 


.9881 


.9886 


.9890 


.9894 


•9899 


•99°3 


.9908 





1 


1 


2 


2 


3 


3 


4 


4 


98 


.991a 


.9917 


.9921 


.9926 


•9930 


•9934 


•9939 


•9943 


•9948 


•9952 





1 


1 


2 


2 


3 


3 


4 


4 


99 -995^ 


.9961 


•9965 


.9969 


•9974 


•9978 


•9983 


•9987 


.9991 


.9996 





1 


1 


2 


2 


3 


3 


3 


4 



632 



SURVEYING. 



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TABLES. 



633 



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640 in 



SURVEYING. 



Z*> 






TABLE V. 
Horizontal Distances and Elevations from Stadia Readings. 












10 


1 





2° 


S 


>° 


Minutes. 




















Hor. 


Diff. 


Hor. 


Diff. 


Hor. 


Diff. 


Hor. 


Diff. 




Dist. 


Elev. 


Dist. 


Elev. 


Dist. 


Elev. 


Dist. 


Elev. 


. . 


100.00 


O.00 


99-97 


I.74 


99.88 


3-49 


99-73 


5-23 


2 , 






« 


O.06 


II 


I.80 


99.87 


3-55 


99.72 


5.28 


4 






i< 


0.12 


(1 


1.86 


11 


3.60 


99.71 


5-34 


6 . 






« 


0.17 


99.96 


I.92 


11 


3.66 


«i 


5-40 


8 . 






« 


O.23 


II 


I.98 


99.86 


3-72 


9970 


5.46 


10 . 






« 


O.29 


(1 


2.04 


11 


3-78 


99.69 


5-52 


12 . 






(« 


°-35 


l< 


2.09 


99-85 


3.84 


11 


5-57 


14 . 






M 


0.41 


99-95 


2.15 


11 


3-90 


99.68 


5-63 


16 






« 


0.47 


(i 


2.21 


99.84 


3-95 


11 


5-69 


18 






H 


0.52 


<< 


2.27 


11 


4.01 


99.67 


5-75 


20 






« 


0.58 


«i 


2-33 


99-83 


4-07 


99.66 


5-8o 


22 






«« 


0.64 


99.94 


2.38 


<i 


4-13 


<i 


5.86 


24 






H 


0.70 


<i 


2.44 


99.82 


4.18 


99-65 


5-92 


26 






99.99 


0.76 


«« 


2.5O 


11 


4.24 


99.64 


5.98 


28 






(( 


0.81 


99-93 


2.56 


99.81 


4.30 


99-63 


6.04 


30 






l< 


0.87 


«< 


2.62 


<i 


4-36 


11 


6.09 


32 






(C 


°-93 


« 


2.67 


99.80 


4.42 


99.62 


6.15 


34 






« 


0.99 


« 


2-73 


it 


4.48 


11 


6.21 


36 






«( 


1.05 


99.92 


2.79 


99-79 


4-53 


99.61 


6.27 


38 






(( 


1. 11 


«< 


2.85 


11 


4-59 


99.60 


6-33 


40 






« 


1.16 


11 


2.9I 


99.78 


4.65 


99-59 


6.38 


42 






t« 


1.22 


99.91 


2.97 


«i 


4.71 


11 


6.44 


44 






99.98 


1.28 


« 


3.02 


99-77 


4.76 


99.58 


6.50 


46 






(« 


i.34 


99.90 


3.08 


11 


4.82 


99-57 


6.56 


48 






11 


1.40 


<« 


3-14 


99.76 


4.88 


99-56 


6.61 


5° 






a 


i-45 


« 


3.20 


<« 


4.94 


<i 


6.67 


52 






it 


1.51 


99.89 


3.26 


99-75 


4.99 


99-55 


6-73 


54 






l< 


i-57 


11 


3-3 1 


99-74 


5-05 


99-54 


6.78 


56 






99-97 


1.63 


k 


3-37 


11 


5-u 


99-53 


6.84 


58 






«< 


1.69 


99.88 


3-43 


99-73 


5-i7 


99-52 


6.90 


60 






» 


1.74 


<< 


3-49 


11 


5-23 


99-51 


6.96 


c = 1.00 


o-75 


0.01 


0.75 


0.02 


075 


0.03 


0-75 


0.05 


1. 00 


0.01 


1. 00 


0.03 


1. 00 


0.04 


1. 00 


0.06 


r = ] 


.2\ 




125 


0.02 


1.25 


0.03 


1.25 


0.05. 


1.25 


0.08 



■.'■'*' * This table was computed by Mr. Arthur Winslow of the State Geological Survey of Pennsylvania. 



?' 



<f\ 



TABLES. 



64I 



TABLE V '.—Continued. 
Horizontal Distances and Elevations from Stadia Readings. 







4 





5 





6° 


7 


1 


Minutes. 




















Hor. 


Diff. 


Hor. 


Diff. 


Hor. 


Diff. 


Hor. 


Diff. 




Dist. 


Elev. 


Dist. 


Elev. 


Dist, 


Elev. 


Dist. 


Elev. 


O . . 


99-5 1 


6.96 


99.24 


8.68 


98.91 


IO.40 


98.51 


12.10 


2 . 




(« 


7-02 


99-23 


8.74 


98.90 


IO.45 


98.50 


12.15 


4 • 




99-5° 


7.07 


99.22 


8.80 


98.88 


IO.51 


98.48 


12.21 


6 . 




99.49 


7-*3 


99.21 


8.85 


98.87 


10-57 


98.47 


12 26 


8 - 




99.48 


7.19 


99.20 


8.91 


98.86 


IO.62 


98.46 


I-.32 


10 . 




99-47 


7-25 


99.19 


8.97 


98.85 


IO.68 


98.44 


" 12.38 


12 




99.46 


7-30 


99.18 


9-03 


98.83 


IO.74 


98-43 


12.43 


14 . 




<( 


7-36 


99.17 


9.08 


98.82 


IO.79 


98.4I 


12.49 


16 . 




9945 


7.42 


99.16 


9.14 


98.81 


IO.85 


98.40 


12-55 


18 




99.44 


7.48 


99- 1 5 


9.20 


98.80 


IO.9I 


98.39 


12.60 


20 




9943 


7-53 


99.14 


9- 2 5 


98.78 


IO.96 


98.37 


12.66 


22 




99.42 


7-59 


99-13 


9-3 1 


98.77 


II.02 


98.36 


12.72 


24 




99.41 


7-65 


99.11 


9-37 


98.76 


II.08 


98.34 


12.77 


26 




99.40 


7.71 


99.10 


9-43 


98.74 


II. 13 


98.33 


12.83 


28 




99-39 


7.76 


99.09 


9.48 


98.73 


II. 19 


98.31 


12.88 


30 




j 99-38 


7.82 


99.08 


9-54 


98.72 


II.25 


98.29 


12.94 


32 




99-38 


7.88 


99.07 


9.60 


98.71 


II.30 


98.28 


13.00 


34 




99-37 


7-94 


99.06 


9-65 


98.69 


II.36 


98.27 


x 3-o5 


36 




99-36 


7-99 


99.05 


9.71 


98.68 


II.42 


98.25 


13.11 


38 




99-35 


8.05 


99.04 


9-77 


98.67 


II.47 


98.24 


I3-I7 


40 




99-34 


8.1 1 


99-03 


9-83 


98.65 


n-53 


98.22 


13.22 


42 




99-33 


8.17 


99.01 


9.88 


98.64 


11.59 


98.20 


13.28 


44 




99-32 


8.22 


9900 


9.94 


98.63 


11.64 


98.19 


13-33 


46 




99-3 1 


8.28 


98.99 


10.00 


98.61 


11.70 


98.17 


»3-39 


48 




99-30 


8.34 


98.98 


10.05 


98.60 


11.76 


98.16 


1345 


50 




99.29 


8.40 


98.97 


10.11 


98.58 


11.81 


98.I4 


I3-50 


52 




99.28 


8-45 


98.96 


10.17 


98.57 


11.87 


98.13 


I3-56 


54 




99.27 


8.51 


98.94 


10.22 


98.56 


n-93 


98.II 


13.61 


56 




99.26 


8.57 


98-93 


10.28 


98-54 


11.98 


98.IO 


13-67 


58 




99.25 


8.63 


98.92 


10.34 


98.53 


12.04 


98.08 


*3-73 


60 




99.24 


8.68 


98.91 


10.40 


98.51 
O.75 


12.10 


98.06 


I3-78 


' = 0.75 

c — 1 .00 


0.75 


0.06 


0.75 


0.07 


0.08 


0-74 


0.10 


1. 00 


0.08 


0.99 


0.09 


O.99 


0.1 1 


O.99 


0.13 


<r = 


[.25 


1.25 


0.10 


1.24 


0.1 1 


I.24 


0.14 


I.24 


0.16 



c 






642 



SUR VE YING. 



TABLE V —Continued. 
Horizontal Disiances and Elevations from Stadia Readings. 







s 





S 


>° 


io° 


„ 

11° 


Minutes. 




















Hor. 


Diff. 


Hor. 


Diff. 


Hor. 


Diff. 


Hor. 


Diff. 




Dist. 


Elev. 


Dist. 


Elev. 


Dist. 


Elev. 


Dist. 


Elev. 


O . . 


98.06 


13.78 


97-55 


1545 


96.98 


17.IO 


96.36 


18.73 


2 




98.05 


13.84 


97-53 


I5-5I 


96.96 


17.16 


96.34 


18.78 


4 




98.03 


I3.89 


97-52 


15.56 


96.94 


17.21 


96.32 


18.84 


6 




98.OI 


13-95 


97-50 


15.62 


96.92 


17.26 


96.29 


18.89 


8 




98.OO 


I4.OI 


97.48 


I5- 6 7 


96.90 


I7-32 


96.27 


18.95 


10 




97.98 


14.06 


97.46 


15-73 


96.88 


17-37 


96.25 


19.OO 


12 




97-97 


14.12 


9744 


15.78 


96.86 


1743 


96.23 


19.05 


14 




97-95 


14.17 


97-43 


15.84 


96.84 


17.48 


96.21 


19.II 


16 




97-93 


14.23 


97.41 


15.89 


96.82 


17-54 


96.18 


19.16 


18 




97.92 


14.28 


97-39 


: 5-95 


96.80 


T 7-59 


96.16 


19.21 


20 




97.90 


14-34 


97-37 


16.00 


96.78 


17.65 


96.14 


19.27 


22 




97.88 


14.40 


97-35 


16.06 


96.76 


17.70 


96.12 


19.32 


24 




97.87 


14-45 


97-33 


16.11 


96.74 


17.76 


96.09 


19.38 


26 




97.85 


14-51 


97-3 1 


16.17 


96.72 


17.81 


96.07 


1943 


28 




97.83 


14.56 


97.29 


16.22 


96.70 


17.86 


96.05 


19.48 


30 




97.82 


14.62 


97.28 


16.28 


96.68 


17.92 


96.03 


19-54 


32 




97.80 


14.67 


97.26 


rf-33 


96.66 


17.97 


96.OO 


19-59 


34 




97.78 


14-73 


97.24 


16.39 


96.64 


18.03 


95-98 


19.64 


36 




97.76 


14.79 


97.22 


16.44 


96.62 


18.08 


95-96 


19.70 


38 




97-75 


14.84 


97-20 


16.50 


96.60 


18.14 


95-93 


1975 


40 




97-73 


14.90 


97.18 


16.55 


96.57 


18.19 


95-91 


19.80 


42 




97.71 


14-95 


97.16 


16.61 


96.55 


18.24 


95-89 


19.86 


44 




97.69 


15.01 


97-14 


16.66 


96-53 


18.30 


95.86 


19.91 


46 




97.68 


15.06 


97.12 


16.72 


96.51 


18.35 


95-84 


19.96 


48 




97.66 


15.12 


97.10 


16.77 


96.49 


18.41 


95.82 


20.02 


50 




97.64 


15-17 


97.08 


16.83 


96.47 


18.46 


95-79 


20.07 


52 




97.62 


r 5- 2 3 


97.06 


16.88 


96-45 


18.51 


95-77 


20.12 


54 




97.61 


15.28 


97.04 


16.94 


96.42 


18.57 


95-75 


20.l8 


56 




97-59 


15-34 


97.02 


16.99 


96.40 


18.62 


95-72 


20.23 


58 




97-57 


I5-40 


97.00 


17-05 


96-38 


18.68 


95-70 


20.28 


60 




97-55 


15-45 


96.98 


17.10 


96.36 


18.73 


95.68 


20.34 


c = o.75 
c — 1.00 


0.74 


0.1 1 


0.74 


0.12 


O.74 


0.14 


073 


O.I5 


0.99 


0.15 


0.99 


0.16 


O.98 


0.18 


0.98 


0.20 


C = I 


•25 


1.23 


0.18 


1.23 


0.21 


I.23 


0.23 


1.22 


O.25 



TABLES. 



643 



TABLE V .—Continued. 
Horizontal Distances and Elevations from Stadia Readings. 



Minutes, 


11 


2° 


13° 


14° 


15° 




















Hor. 


Diff. 


Hor. 


Diff. 


Hor. 


Diff. 


Hor. 


Diff. 




Dist. 


Elev. 


Dist. 


Elev. 


Dist. 


Elev. 


Dist. 


Elev. 


O . . 


95.6S 


20.34 


94.94 


21.92 


94-15 


23-47 


93-30 


25.OO 


2 . . 


95.65 


20.39 


94.91 


21.97 


94.12 


23-52 


93-27 


25.05 


4 • • 


95-63 


20.44 


94.89 


22.02 


94.09 


23-58 


93-24 


25.IO 


6 . . 


95.61 


20.50 


94.86 


22.o8 


94.07 


23-63 


93.21 


2 5- T 5 


8 . . 


95-58 


20-55 


94.84 


22.13 


94.04 


23.68 


93.18 


25.20 


10 . . 


95-56 


20.60 


94.81 


22.l8 


94.OI 


23-73 


93.16 


25-25 


12 . . 


95-53 


20.66 


94-79 


22.23 


93-98 


23.78 


93-13 


25-30 


14 . . 


95-5* 


20.71 


94.76 


22.28 


93-95 


23-83 


93.10 


25-35 


16 . . 


95-49 


20.76 


94-73 


22.34 


93-93 


23.88 


93-°7 


25.40 


18 . . 


95-46 


20.81 


94.71 


22.39 


93-9o 


23-93 


93-°4 


2545 


20 . . 


95-44 


20.87 


94.68 


22.44 


93-87 


23-99 


93.01 


25-50 


22 . . 


95-41 


20.92 


94.66 


22.49 


93-84 


24.04 


92.98 


25-55 


24 . . 


95-39 


20.97 


94-63 


22.54 


93.81 


24.09 


92.95 


25.60 


20 . . 


95-36 


21.03 


94.60 


22.6o 


93-79 


24.14 


92.92 


25.65 


28 . . 


j 95-34 


21.08 


94.58 


22.65 


93-76 


24.19 


92.89 


25.70 


30 . . 


| 95-32 


21.13 


94-55 


22.70 


93-73 


24.24 


92.86 


2575 


32 . . 


95-29 


21.18 


94.52 


22.75 


93-70 


24.29 


92.83 


25.80 


34 • • 


95.27 


21.24 


94-5o 


22.8o 


93-67 


24-34 


92.80 


25-85 


3 6 . . 


95-24 


21.29 


94-47 


22.85 


93-65 


24-39 


92.77 


25.90 


38 • • 


95.22 


21.34 


94-44 


22.91 


93.62 


24.44 


92.74 


25-95 


40 . . 


95-19 


21.39 


94.42 


22.96 


93-59 


24.49 


92.71 


26.00 


42 . . 


95-17 


21.45 


94-39 


23.OI 


93-56 


24-55 


92.68 


26.05 


44 • • 


95-14 


21.50 


94-36 


23.06 


93-53 


24.60 


92.65 


26.10 


46 . . 


95-12 


2I -55 


94-34 


23.II 


93-5° 


24.65 


92.62 


26.15 


48 . . 


95.09 


21.60 


94-31 


23.16 


93-47 


24.70 


92-59 


26.20 


50 . . 


95.07 


21.66 


94.28 


23.22 


93-45 


24-75 


92.56 


26.25 


52 . . 


95-Q4 


21.71 


94.26 


23.27 


93-42 


24.80 


92-53 


26.30 


54 • • 


95.02 


21.76 


94-23 


23-32 


93-39 


24.85 


92.49 


26.35 


56 . . 


94.99 


21.81 


94.20 


2 3-37 


93-36 


24.90 


92.46 


26.40 


58 . . 


94-97 


21.87 


94-17 


23.42 


93-33 


24-95 


92-43 


26.45 


60 . . 
'=0.75 
c = 1.00 
c — 1.25 


94.94 


21.92 


94-15 


23-47 


93-3o 


25.00 


92.40 


26.50 


o-73 


0.16 


o-73 


0.17 


o-73 


0.19 


0.72 


0.20 


0.98 


0.22 


0.97 


0.23 


0.97 


0.25 


0.96 


0.27 


1.22 

1 


0.27 


1.21 


0.29 


1. 21 


0.31 


1.20 


o-34 



644 



SURVEYING. 



TABLE V. — Continued. 
Horizontal Distances and Elevations from Stadia Readings. 







16° 


17° 


18° 


19° 


Minutes. 




















Hor. 


Diff. 


Hor. 


Diff. 


Hor. 


Diff. 


Hor. 


Diff 




Dist. 


Elev. 


Dist. 


Elev. 


Dist. 


Elev. 


Dist. 


Elev. 


. . 


92.40 


26.50 


91-45 


27.96 


90-45 


29-39 


89.40 


30.78 


2 




92-37 


26.55 


91.42 


28.OI 


90.42 


29.44 


89.36 


30.83 


4 




92.34 


26.59 


9^39 


28.06 


90.38 


29.48 


89-33 


30.87 


6 




92.31 


26.64 


9 r 35 


2S.IO 


90-35 


29-53 


89.29 


30.92 


8 




92.28 


26.69 


91.32 


28.15 


90.3t 


29.58 


89.26 


30.97 


10 




92.25 


26.74 


91.29 


28.20 


90.28 


29.62 


89.22 


31.OI 


12 




92.22 


26.79 


91.26 


28.25 


90.24 


29.67 


89.18 


31.06 


14 




92.19 


26.84 


91.22 


28.30 


90.21 


29.72 


89.15 


31.IO 


16 




92.15 


26.S9 


91.19 


28.34 


90.18 


29.76 


89.II 


3I-I5 


18 




92.12 


26.94 


91.16 


28.39 


90.14 


29.81 


89.08 


3r-I9 


20 




92.09 


26.99 


91.12 


28.44 


90.11 


29.86 


89.04 


3!- 2 4 


22 




92.06 


27.04 


91.09 


28.49 


90.07 


29.90 


89.OO 


31.28 


24 




92.03 


27.09 


91.06 


28.54 


90.04 


29-95 


88.96 


31.33 


26 




92.OO 


27-I3 


91.02 


2S.5S 


90.OO 


30.00 


88.93 


V-3* 


28 




91.97 


27.18 


90.99 


2S.63 


S9-97 


30.04 


S8.89 


31.42 


30 




91-93 


27.23 


90.96 


2S.68 


89-93 


30.09 


88.86 


3*47 


32 




91.90 


27.2S 


90.92 


2S.73 


89.90 


30.14 


88.82 


31-51 


34 




91.87 


27-33 


90.89 


2S.77 


89.86 


30.19 


88.78 


3I-56 


36 




91.84 


27.38 


90.86 


2S.82 


89.83 


30.23 


88.75 


31.60 


38 




91.81 


27-43 


90.82 


28.S7 


89.79 


30.28 


88.71 


31-65 


40 




9 r -77 


27.48 


90.79 


2S.92 


89.76 


30.32 


88.67 


31.69 


42 


91.74 


27.52 


90.76 


2S.96 


89.72 


30-37 


88.64 


3 T -74 


44 • • 


91.71 


27.57 


90.72 


29.OI 


89.69 


30.41 


88.60 


31-78 


46 . . 


91.68 


27.62 


90.69 


29.06 


89.65 


30.46 


88.56 


3I-83 


48 . . 


91.65 


27.67 


90.66 


29.II 


8961 


30-51 


83.53 


31-87 


50 - . 


91.61 


27.72 


90.62 


29-I5 


89.58 


30-55 


88.49 


31.92 


52 • • 


91.58 


27.77 


90-59 


29.20 


89.54 


30.60 


88.45 


31.96 


54 ■ • 


91-55 


27.81 


90-55 


29.25 


89.51 


30-65 


88.41 


32.01 


5 6 . . 


91.52 


27.S6 


90.52 


29.30 


89.47 


30.69 


88.38 


32.05 


58 . . 


91.48 


27.91 


90.48 


29-34 


89.44 


3074 


8S.34 


32.09 


60 . . 

' = 0.75 

c = 1.00 


91.45 


27.96 


90.45 


29-39 


89.40 
O.71 


30.78 


88.30 


32.14 


0.72 


0.2I 


0.72 


0.23 


0.24 


0.71 


0.25 


0.86 


O.28 


o-95 


0.30 


o-95 


0.32 


0-94 


o.33 


1.20 


o-35 


1. 19 


0.38 


1. 19 


0.40 


1.18 


0.42 






1 










J 


1 1 



TABLES. 



645 



TABLE V .—Continued. 
Horizontal Distances and Elevations from Stadia Readings. 



Minutes. 



o 

2 

4 

6 

8 

10 

12 

14 

16 
18 
20 

22 

24 
26 
28 
3° 



20 c 



Hor. 
Dist. 



88.30 
88.26 
88.23 
88.19 
88.15 

88.11 

88.08 
88.04 
88.00 
87.96 
87.93 

87.89 
87.85 
87.81 
8777 
87-74 



Diff. 
Elev. 



3 2 - J 4 

32.18 

3 2 - 2 3 
32.27 

32-3 2 
32-36 



21< 



Hor. 
Dist. 



87.16 
87.12 
87.08 
87.04 
87.OO 
86.96 



Diff. 
Elev. 



32.41 


86.92 


3245 


86.88 


3249 


86.84 


32.54 


86.80 


32.58 


86.77 


32.63 


86.73 


32.67 


86.69 


32.72 


86.65 


32.76 


86.61 


32.S0 


86.57 



22° 



23< 



n 



Hor. 
Dist. 



33-46 
33-50 

33-54 
33-59 
33-63 
33-67 

33-72 
33-76 
33-8o 
33-84 
33-89 

33-93 
33-97 
34.01 
34.06 
34.10 



Diff. 

Elev. 



85-97 

85-93 
85.89 
85.85 
85.S0 
85.76 

85.72 
85.68 
85.64 
85.60 
85.56 

85.52 

85.48 

85-44 
8540 
85.36 



34-73 
34-77 
34.82 
34.86 
34-90 
34-94 

34-98 
35.02 

35-°7 
35- rI 
35-15 

35-*9 
35-23 
35-27 
35-31 
35-36 



Hor. 
Dist. 



84-73 
84.69 
84.65 
84-61 
84.57 
84.52 

8448 

84-44 
84.40 

84.35 
84.31 

84.27 
84.23 
84.18 

84.14 
84.IO 

84.06 
84.OI 
83-97 
83-93 
83.89 



Diff. 
Elev. 



35-97 

36.01 

36-05 
36.09 

36.13 
36.17 

36.21 

36.25 

36.29 

36-33 
36-37 

36.41 
3645 
36-49 
36-53 
36-57 

36.61 

36-65 
36.69 

36.73 
36-77 




646 



SURVEYING. 



TABLE V .—Continued. 
Horizontal Distances and Elevations from Stadia Readings. 





24° 


25° 


26° 


27° 


Minutes. 




















Hor. 


Diff. 


Hor. 


Diff. 


Hor. 


Diff. 


Hor. 


Diff. 




Dist 


Elev. 


Dist. 


Elev 


Dist. 


Elev 


Dist. 


Elev. 


O . . 


83.46 


37.16 


82.14 


38.30 


80.78 


39-40 


79-39 


40.45 


2 . . 


83.41 


37.20 


82.09 


38.34 


80.74 


39-44 


79-34 


40.49 


4 • • 


83-37 


37.23 


82.05 


38.38 


80.69 


39-47 


79-30 


40.52 


6 . . 


83-33 


37-27 


82.OI 


38.41 


80.65 


39-5 1 


79-25 


40.55 


8 . . 


83.28 


37-3 1 


81.96 


3845 


80.60 


39-54 


79.20 


40.59 


10 . . 


83.24 


37-35 


81.92 


38.49 


80.55 


39-58 


79-15 


40.62 


12 . . 


83.20 


37-39 


81.87 


38-53 


80.51 


39.61 


79.11 


40.66 


14 . . 


83-15 


37-43 


81.83 


38.56 


80.46 


39-65 


79.06 


40.69 


16 . . 


83.11 


37-47 


81.78 


38.60 


80.41 


39-69 


79.01 


40.72 


18 . . 


83.07 


37-51 


81.74 


38.64 


80.37 


39-72 


78.96 


40.76 


20 . . 


83.02 


37-54 


81.69 


38.67 


80.32 


39-76 


78.92 


40.79 


22 . . 


82.98 


37-58 


81.65 


38.71 


80.28 


39-79 


78.87 


40.82 


24 . . 


82.93 


37.62 


8l.6o 


38.75 


80.23 


39-83 


78.82 


40.86 


26 . . 


82.89 


37.66 


81.56 


3878 


80.18 


39.86 


78.77 


40.89 


28 . . 


82.85 


37-70 


81.51 


38.62 


80.14 


39-9o 


78.73 


40.92 


30 . . 


82.80 


37-74 


81.47 


38.86 


80.09 


39-93 


78.68 


40.96 


32 . . 


82.76 


37-77 


81.42 


38.89 


80.04 


39-97 


78.63 


40.99 


34 • • 


82.72 


37-81 


81.38 


• 38-93 


80.OO 


40.00 


78.58 


41.02 


36 • . 


82.67 


37-85 


8i-33 


38.97 


79-95 


40.04 


78.54 


41.06 


38 . . 


82.63 


37-89 


81.28 


39.OO 


79.90 


40.07 


78.49 


41.09 


40 . . 


82.58 


37-93 


81.24 


39-04 


79.86 


40.11 


78.44 


41.12 


42 . . 


82.54 


37-96 


81.19 


39.08 


7981 


40.14 


78.39 


41.16 


44 • • 


82.49 


38.00 


81.15 


39-" 


79 76 


40.18 


78.34 


41.19 


46 . . 


82.45 


38.04 


81.10 


39- 1 5 


79-72 


40.21 


78.30 


41.22 


48 . . 


82.41 


38.08 


81.06 


39.18 


79.67 


40.24 


7825 


41.26 


50 • . 


82.36 


38.11 


81.01 


39.22 


79.62 


40.28 


78.20 


41.29 


52 • • 


82.32 


38-15 


80.97 


39.26 


79-58 


40.31 


78.15 


41.32 


54 • - 


82.27 


38.19 


80.92 


39-29 


79-53 


40.35 


78.ro 


,4L35 


56 . . 


82.23 


38-23 


80.87 


39-33 


79.48 


40.38 


78.06 


41.39 


58 . . 


82.18 


38.26 


80.83 


39-36 


79-44 


40.42 


78.01 


41.42 


60 . . 

' = 0.75 

c = 1 .00 
<: = I.25 


82.14 


38-30 


80.78 


39-4Q 


79-39 


40.45 


77.96 


41.45 


0.68 


0.31 


0.68 


0.32 


0.67 


o.33' 


0.66 


o-35 


0.91 


0.41 


0.90 


0.43 


0.89 


0.45 


0.89 


0.46 


1.14 


0.52 


"3 


0.54 


1. 12 


0.56 


1. 11 


0.58 



TABLES. 



647 



TABLE V .—Continued. 
Horizontal Distances and Elevations from Stadia Readings. 





28° 


29° 


30° 


Minutes. 
















Hor. 


Diff. 


Hor. 


Diff. 


Hor. 


Diff. 




Dist. 


Elev. 


Dist 


Elev. 


Dist. 


Elev. 


. . 


77.96 


4i-45 


76.50 


42.40 


75.OO 


43-30 


2 . 




77.91 


41.48 


76.45 


4243 


74-95 


43-33 


4 • 




77.86 


41.52 


76.40 


42.46 


74.90 


43-36 


6 . 




77.81 


41-55 


76.35 


4249 


74.85 


43-39 


8 . 




7777 


41.58 


76.30 


42.53 


74.S0 


43-42 


10 . 




77.72 


41.61 


76.25 


42.56 


74-75 


43-45 


12 . 




77.67 


41.65 


76.20 


42.59 


74.70 


43-47 


14 . 




77.62 


41.68 


76.15 


42.62 


74.65 


43-50 


16 . 




77-57 


41.71 


76.IO 


42.65 


74.60 


43-53 


18 . 




77.52 


41.74 


76.05 


42.68 


74-55 


43-56 


20 . 




77.48 


4^-77 


76.OO 


42.71 


74-49 


43-59 


22 




77.42 


41.81 


75-95 


42.74 


74-44 


43.62 


24 




77.38 


41.84 


75-9° 


42.77 


74-39 


43-65 


26 . 




77-33 


41.87 


75-85 


42.80 


74-34 


43-67 


28 




77.28 


41.90 


75.80 


42.83 


74.29 


43-7o 


30 




77-23 


fi-93 


75-75 


42.86 


74.24 


43-73 


32 




77.18 


41.97 


75-70 


42.89 


74.19 


43-76 


34 




77-13 


42.00 


75-65 


42.92 


74.14 


43-79 


36 




77.09 


42.03 


75.60 


42.95 


74.09 


43.82 


38 




77.04 


42.06 


75-55 


42.9S 


74.04 


43-84 


40 




76.99 


42.09 


75-5o 


43.OI 


73-99 


43-87 


42. 




76.94 


42.12 


75-45 


43-04 


73-93 


43-90 


44 




76.89 


42.15 


7540 


43-°7 


73.88 


43-93 


46 




76.84 


42.19 


75-35 


43.10 


73-83 


43-95 


48 




76.79 


42.22 


75-3° 


43- T 3 


73-78 


43-98 


5° 




76.74 


42.25 


75-25 


43.16 


73-73 


44.01 


52 




76.69 


42.28 


75.20 


43.18 


73.68 


44.04 


54 




76.64 


42.31 


75-15 


43-21 


73-63 


44.07 


56 




76.59 


42.34 


75- 10 


43-24 


73-58 


44.09 


58 




76.55 


42.37 


75-°5 


43-27 


73-52 


44.12 


60 




76.50 


42.40 


75.00 


43-30 


73-47 


44-15 


c = o.75 

c— 1.00 


0.66 


0.36 


0.65 


0-37 


0.65 


0.38 


0.88 


0.48 


0.87 


0.49 


0.86 


0.51 


1. 10 


0.60 


1.09 


0.62 


1.08 


0.64 
















1 >•>. 



648 



SURVEYING. 



TABLE VI. 

Natural Sines and Cosines. 






0° 


1° 


2° 


3° 


40 


60 


Sine 

Tooooo 


Cosin 


Sine 
.01745 


Cosin 


Sine j Cosin 
.03490 .99939 


Sine 
.05234 


Cosin 


Sine 


Cosin 
.99756 


One. 


.99985 


.99863 


.06976 


1 


.00029 


One. 


.01774 


.99984 


.03519 


.99938 


.05263 


.99861 


.07005 


.99754 


59 


2 


.00058 


One. 


.01803 


.99984 


.03548 


.99937 


.05292 


.99860 


.07034 


.99752 


58 


3 


.00087 


One. 


.01832 


.99983 


.03577 


.99936 


.05321 


.99858 


.07063 


.99750 


57 


4 


.00116 


One. 


.01862 


.99983 


.03606 


.99935 


.05350 


.99857 


.07092 


.99748 


56 


5 


.00145 


One. 


.01891 


.99982 


.03635 


.99934 


.05379 


.99855 


.07121 


.99746 


55 


6 


.00175 


One. 


.01920 


.99982 


.03664 


.99933 


.05408 


.99854 


.07150 


.99744 


54 


7 


.00204 


One. 


.01949 


.99981 


.03693 


.99932 


.05437 


.99852 


.07179 


.99742 


53 


8 


.00233 


One. 


.01978 


.99980 


.03723 


.99931 


.05466 


.99851 


.07208 


.99740 


52 


9 


.00262 


One. 


.02007 


.99980 


.03752 


.99930 


.05495 


.99849! 


.07237 


.99738 


51 


10 


.00291 


One. 


.02036 


.99979 


.03781 


.99929 


.05524 


.99847 


.07266 


.99736 


50 


11 


.00320 


.99999 


.02065 


.99979 


.03810 


.99927 


.05553 


.99846 


.07295 


.99734 


49 


12 


.00349 


.99999 


.02094 


.99978 


.03839 


.99926 


.05582 


.99844 


.07324 


.99731 


48 


13 


.00378 


.99999' 


.02123 


.99977 


.03868 


.99925 


.05611 


.99842 


.07353 


.99729 


47 


14 


.00407 


. 99999 j 


.02152 


.99977 


.03897 


.99924 


.05640 


.99841 


.07382 


.99727 


46 


15 


.00436 


.99999! 


.02181 


.99976 


.03926 


.99923 


.05669 


.99839 


.07411 


.99725 


45 


16 


.00465 


.99999! 


.02211 


.99976 


.03955 


.99922 


.05698 


.99838 


.07440 


.99723 


44 


17 


.00495 


.99999! 


.02240 


.99975 


.03984 


.99921 


.05727 


.99836 


.07469 


.99721 


43 


18 


.00524 


.99999' 


.02269 


.99974 


.04013 


.99919 


.05756 


.99834 


.07498 


.99719 


42 


19 


.00553 


.99998, 


.02298 


.99974 


.04042 


.99918 


.05785 


.99833 


.07527 


.99716 


41 


20 


.00582 


.99998 


.02327 


.99973 


.04071 


.99917 


.05814 


.99831 


.07556 


.99714 


40 


21 


.00611 


.999981 


.02356 


.99972 


.04100 


.99916 


.05844 


.99829 


.07585 


.99712 


39 


22 


.00640 


.99998! 


.02385 


.99972 


.04129 


.99915 


.05873 


.99827 


.07614 


.99710 


38 


23 


.00669 


.99998 


.02414 


.99971 


.04159 


.99913 


.05902 


.99826 


.07643 


.99708 


37 


24 


.00698 


.99998 


.02443 


.99970 


.04188 


.99912 


.05931 


.99824 


.07672 


.99705 


36 


25 


.00727 


.99997 


.02472 


.99969 


.04217 


.99911 


.05960 


.99822 


.07701 


.99703 


35 


26 


.00756 


.99997 


.02501 


.99969 


.04246 


.99910 


.05989 


.99821 


.07730 


.99701 


34 


27 


.00785 


.99997 


.02530 


.99968 


.04275 


.99909 


.06018 


.99819 


.07759 


.99699 


33 


28 


.00814 


.99997 


.02560 


.99967 


.04304 


.99907 


.06047 


.99817 


.07788 


.99696 


32 


29 


.00844 


.99996 


.02589 


.99966 


.04333 


.99906 


.06076 


.99815 


.07817 


.99694 


31 


30 


.00873 


.99996 


.02618 


.99966 


.04362 


.99905 


.06105 


..99813 
". 99812 


.07846 


.99692 


30 


31 


.00902 


.99996 


.02647 


.99965 


.04391 


.99904 


.06134 


.07875 


.99689 


29 


32 


.00931 


.99996 


.02676 


.99964 


.04420 


.99902 


.06163 


.99810 


.07904 


.99687 


28 


33 


.00960 


.99995; 


.02705 


.99963 


.04449 


.99901 


06192 


.99808 


.07933 


.99685 


27 


34 


.00989 


.99995, 


.02734 


.99903 


.04478 


.99900 


.06221 


.99806 


.07962 


.99683 


26 


35 


.01018 


.999951 


.02763 


.99962 


.04507 


.99898 


.06250 


.99804 


.07991 


.99680 


25 


36 


.01047 


.99995 


.02792 


.99961 


.04536 


.99897 


.00279 


.99803 


.08020 


.99678 


24 


37 


.01076 


.99994 


.02821 


.99960 


.04565 


.99896 


.06308 


.99801 


.08049 


.99676 


23 


38 


.01105 


.99994! 


.02850 


.99959 


.04594 


.99894 


.06337 


.99799 


.08078 


.99673 


22 


39 


.01134 


.99994 


.02879 


.99959 


.04623 


.99893 


.00366 


.99797 


.08107 


.99671 


21 


40 


.01164 


.99993 


.02908 


.99958 


.04653 


.99892 


.06395 


.99795 


.08136 


.99668 


20 


41 


.01193 


.99993 


.02938 


.99957 


.04682 


.99890 


.06424 


.99793 


.08165 


.99666 


19 


42 


.01222 


.99993 


.02967 


.99956 


.04711 


.99889 


.00453 


.99792 


.08194 


.99664 


18 


43 


.01251 


.99992 


.02996 


.99955 


.04740 


.99888 


.06482 


.99790 


.08223 


.99661 


17 


44 


.01280 


.99992 


.03025 


.99954 


.04769 


.99886 


.06511 


.99788 


.08252 


.99659 


16 


45 


.01309 


.99991 


.03054 


.99953 


.04798 


.99885 


.06540 


.99786 


.08281 


.99657 


15 


46 


.01338 


.99991 


.03083 


.99952 


.04827 


.99883 


.06569 


.99784 


.08310 


.99654 


14 


47 


.01367 


.99991 


.03112 


.99952 


.04856 


.99882 


.06598 


.99782 


.08339 


.99652 


13 


48 


.01396 


.99990 


.03141 


.99951 


.04885 


.99881 


.06627 


.99780 


.08368 


.99649 


12 


49 


.01425 


.99990 


.03170 


.99950 


.04914 


.99879 


.06656 


.99778 


.08397 


.99647 


11 


50 


.01454 


.99989 


.03199 


.99949 


.04943 


.99878 


.06685 


.99776 


.08426 


.99644 


10 


51 


.01483 


.99989 


.03228 


.99948 


.04972 


.99876 


.06714 


.99774 


.08455 


.99642 


9 


52 


.01513 


.99989 


.032.-7 


.99947 


.05001 


.99875 


.06743 


.99772 


.08484 


.99639 


8 


53 


.01542 


.99988 


.03286 


.99946 


.05030 


.99873 


.06773 


.99770 


.08513 


.99637 


7 


54 


.01571 


.99988 


.03316 


.99945 


.05059 


.99872 


.06802 


.99768 


.08542 


.99635 


6 


55 


.01600 


.99987 


.03345 


.99944 


.05088 


.99870 


.06831 


.99766 


.08571 


.99632 


5 


56 


.01629 


.99987 


.03374 


.99943 


.05117 


.99869 


.06860 


.99764 


.08600 


.99630 


4 


57 


.01658 


.99986 


.03403 


.99942 


.05146 


.99867 


.06889 


.99762 


.08629 


.99627 


3 


58 


.01687 


.99986 


.03432 


.99941 


.05175 


.99866 


.06918 


.99760 


.08658 


.99625 


2 


59 


.01716 


.99985 


.03461 


.99940 


.05205 


.99864 


.06947 


.99758! 


.08687 


.99622 


1 


60 

/ 


.01745 


.99985 


.03490 
Cosin 


.99939 


.05234 


.99863 


.06976 
Cosin 


.99756 


.08716 


.99619 


_0 

/ 


Cosin | Sine 


Sine 


Cosin 


Sine 


Sine 


Cosin 


Sine 


89° 


88° 


87° 


86° 


85° 



TABLES. 



649 



TABLE VI.— Continued. 

Natural Sines and Cosines. 





5° 


6° 


7° 


8° 


9° 


60 


Sine 

.08716 


Cosin 


Sine 


Cosin 

799452 


Sine 

.12187 


Cosin 


Sine 
.13917 


Cosin 
.99027 


Sine j 
.156431 


Cosin 
.98769, 


.99619 


.10453 


.99255 


1 


.08745 


.99617 


.10482 


.99449 


.12216 


.99251 


.13946 


.99023 


.156721 


.98764 


59 


2 


.08774 


.99614 


.10511 


.99446 


.12245 


.99248 


.13975 


.99019 


.15701 


.98760 


58 


3 


.08803 


.99612 


.10540 


.99443 


.12274 


.99244 


.14004 


.99015 


.15730 


.98755J 


57 


4 


.08831 


.99609 


.10569 


.99440 


.12302 


.99240! 


.14033 


.99011 


.15758! 


.98751 


56 


5 


.08860 


.99607 


.10597 


.99437 


.12331 


.99237! 


.14061 


.99000 


.157871 


.98746 


55 


6 


.08889 


.99604 


.10626 


.99434 


.12360 


.99233- 


.14090 


.99002 


.15816! 


.987411 


54 


7 


.08918 


99602 


.10055 


.99431 


.12389 


.99230 


.14119 


.98998 


.15845 


.98737' 


53 


8 


.08947 


.99599 


.10684 


.99428 


.12418 


.992261 


.14148! 


.98994 


.15873 


.98732' 


52 


9 


.08976 


.99596 


.10713 


.99424 


.12447 


.99222 


.14177 ' 


.98990 


.15902 


.98728 


51 


10 


.09005 


.99594 


.10742 


.99421 


.12476 


.99219 


.14205 


.98986 


.15931 


. 98723 | 


50 


11 


.09034 


.99591 


.10771 


.99418 


.12504 


.99215 


.14234 


.98982 


.15959 


.98718 


49 


12 


.09063 


.99588 


.10800 


.99415 


.12533 


.99211 


.14263 


.98978 


.15988 


.98714 


48 


13 


.09092 


.99586 


.10829 


.99412 


.12562 


.99208 


.14292 


.98973 


.16017 


.98709 


47 


14 


.09121 


.99583 


.10858 


.99409 


.12591 


.99204| 


.14320 


.98969 


.16046 


.98704 


46 


15 


.09150 


.99580 


.10887 


.99406 


.12620 


.99200 


.14349 


.98965 


.16074 


.98700 


45 


16 


.09179 


.99578 


.10916 


.99402 


.12649 


.99197 


.14378 


.98961 


.16103 


.98695 


44 


17 


.09208 


.99575 


.10945 


.99399 


.12678 


.99193 


.14407 


.98957 


.16132 


.98U90 43 


18 


.09237 


.99572 


.10973 


.99396 


.12706 


.99189 


.14436 


.98953 


.16160 


.986SG 42 


19 


.09266 


.99570 


.11002 


.99393 


.12735 


.99186 


.14464 


.98948 


.16189 


.981581 41 


20 


.09295 


.99567 


.11031 


.99390 


.12764 


.99182 


.14493 


.98944 


.16218 


.98676 40 


21 


.09324 


.99564 


.11060 


.99386 


.12793 


.99178 


.14522 


.98940 


.16246 


.98671 '' 39 


22 


.09353 


.99562 


.11089 


.99383 


.12822 


.99175 


.14551 


.98936 


.16275 


.98667 38 


23 


.09382 


.99559 


.11118 


.99380 


.12851 


.99171 


.14580 


.98931 


.16304 


.98662 37 


24 


.09411 


.99556 


.11147 


.99377 


.12880 


.99167 


.14608 


.98927 


.16333 


.98657 36 


25 


.09440 


.99553 


.11176 


.99374 


.12908 


.99103 


.14637 


.98923 


.16361 


.98652 


35 


26 


.09469 


.99551 


.11205 


.99370 


.12937 


.99160 


.14666 


.98919 


.16390 


.98648 


34 


27 


.09498 


.99548 


.11234 


.99367 


.12966 


.99156 


.14695 


.98914 


1.16419 


.98643 


33 


28 


.09527 


.99545 


.11263 


.99364 


.12995 


.99152 


.14723 


.98910 


.16447 


.98638 


32 


29 


.09556 


.99542 


.11291 


.99360 


.13024 


.99148 


.14752 


.98906 


'16476 


.98633 


31 


30 


.09585 


.99540 


.11320 


.99357 


.13053 


.99144 


.14781 


.98902 


.16505 


.98629 


30 


• 31 


.09614 


.99537 


.11349 


.99354 


.13081 


.99141 


.14810 


.98897 


.16533 


.98624 


29 


32 


.09642 


.99534 


.11378 


.99351 


.13110 


.99137 


.14838 


.98893 


.16562 


.98619 


28 


33 


.09671 


.99531 


.11407 


.99347 


.13139 


.99133 


.14867 


.98889 


.16591 


.98614 


27 


34 


.09700 


.99528 


.11436 


.99344 


.13168 


.99129 


.14896 


.98884 


.16620 


.98609 


26 


35 


.09729 


.99526 


.11465 


.99341 


.13197 


.99125 


.14925 


.98880 


1. 16648 


.98604 


25 


36 


.09758 


.99523 


.11494 


.99337 


.13226 


.99122 


.14954 


.98876 


! . 16677 


.98600 


24 


37 


.09787 


.99520 


.11523 


.99334 


.13254 


.99118 


.14982 


.98871 


.16706 


.98595 


23 


38 


.09816 


.99517 


.11552 


.99331 


.13283 


.99114 


.15011 


.98867 


.16734 


.98590 


22 


39 


.09845 


.99514 


.11580 


.99327 


.13312 


.99110 


.15040 


.98863 


.16763 


.98585 


21 


40 


.09874 


.99511 


.11609 


.99324 


.13341 


.99106 


.15069 


.98858 


.16792 


.98580 


20 


41 


.09903 


.99508 


.11638 


.99320 


.13370 


.99102 


.15097 


.98854 


.16820 


.98575 


19 


42 


.09932 


.99506 


.11667 


.99317 


.13399 


.99098 


.15126 


.98849 


.16849 


.98570 


18 


43 


.09961 


.99503 


.11696 


.99314 


.13427 


.99094 


.15155 


.98845 


.16878 


.98565 


17 


44 


.09990 


.99500 


.11725 


.99310 


.13456 


.99091 


.15184 


.98841 


.16906 


.98561 


It 


45 


.10019 


.99497 


.11754 


•99307 


.13485 


.99087 


.15212 


.98836 


.16935 


.98556 


15 


46 


.10048 


.99494 


.11783 


.99303 


.13514 


.99083 


.15241 


.98832 


.16964 


.98551 


14 


47 


.10077 


.99491 


.11812 


.99300 


.13543 


.99079 


.15270 


.98827 


.16992 


i.98546 


13 


48 


.10106 


.99488 


.11840 


.99297 


.13572 


.99075 


.15299 


.98823 


i .17021 


1.98541 


12 


49 


.10135 


.99485 


.11869 


.99293 


.13600 


.99071 


.15327 


.98818 


! .17050 


j .98536 


JJ 


50 


.10164 


.99482 


.11898 


.99290 


.13629 


.99067 


.15356 


.98814 


.17078 


.98531 


10 


51 


.10192 


.99479 


.11927 


.99286 


.13658 


.99063 


.15385 


.98809 


.17107 


1.98526 


9 


52 


.10221 


.99476 


.11956 


.99283 


.13687 


.99059 


.15414 


.98805 


.17136 


.98521 


8 


53 


.10250 


.99473 


.11985 


.99279 


.13716 


.99055 


.15442 


!. 98800 


.17164 


.98516 


1 


54 


.10279 


.99470 


.12014 


.99276 


.13744 


.99051 


.15471 


!. 98796 


.17193 


1.98511 


1 6 


55 


.10308 


.99467 


j. 12043 


.99272 


.13773 


.99047 


.15500 


t .98791 


i .17222 


j .98506 


5 


56 


.10337 


.99464 


.12071 


.99269 


.13802 


.99043 


.15529 


1.98787 


.17250 


.98501 


4 


57 


10366 


.99461 


! .12100 


.99265 


.13831 


.99039 


.15557 


.98782 


1.17279 


.98496 


3 

1 .-. 


58 


.10395 


.99458 


| .12129 


.99262 


• .13860 


.99035 


.15586 


.98778 


| .17308 


.98491 


1 ? 


59 


.10424 


.99455 


i .12158 


.99258 


! .13889 


.99031 


.15615 


.98773 


.17336 


,.98486 


J 


60 


.10453 


.99452 


.12187 


.99255 


| .13917 


.99027 


.15643 


.98769 


1.17365 


j. 98481 





/ 


Cosin 


| Sine 


Cosin 


Sine 


Cosin 


Sine 


Cosin 


| Sine 


Cosin 


Sine 


/ 


84° 


83° 


82° 


81° 


80° 





650 



SURVEYING. 



TABLE VI.— Continued. 
Natural Sines and Cosines. 





10° 


11° 


12° 


13° 


14° 


60 





Sine Cosin 
. 17365 ! 798481 


Sine 


Cosin 
798163 


Sine 
720791 


Cosin 

.97815 


Sine 
.22495 


Cosin 


Sine 
.24192 


Cosin 


.19081 


.97437 


.97030 


1 


.17393 .98476 


.19109 


.98157 


.20820 


.97809 


.22523 


.97430 


.24220 


.97023 


59 


2 


.17422 .98471 


. 19138 


.98152 


.20848 


.97803 


.22552 


.97424 


.24249 


.97015 


58 


3 


.17451 .98466 


.19167 


.98146 


.20877 


.97797 


.22580 


.97417 


.24277 


.97008 


57 


4 


.17479 .98461 


.19195 


.98140 


.20905 


.97791 


.22608 


.97411 


.24305 


.97001 


56 


5 


.17508 .98455 


.19224 


.98135 


.20933 


.97784 


.22637 


.97404 


! .24333 


.96994 


55 


6 


.17537 .98450 


.19252 


.98129 


.20962 


.97778 


.22665 


-97398 


.24362 


.96987 


54 


7 


.17565 .98445 


.19281 


.98124 


.20990 


.97772 


.22693 


.97391 


.24390 


.96980 


53 


8 


.17591 .98440 


.19309 


.98118! 


.21019 


.97766 


.22722 


.97384 


.24418 


.96973 


52 


9 


.17623 .98435 


.19338 


.98112 ! 


.21047 


.97760 


.22750 


.97378 


.24446 


.96966 


51 


10 


.17651 j. 98430 


.19366 


.98107, 


.21076 


.97754 


.22778 


.97371 


.24474 


.96959 


50 


11 


.17680 


.98425 


.19395 


.98101' 


.21104 


.97748 ! 


.22807 


.97365 


.24503 


.96952 


49 


12 


.17708 


.98420 


.19423 


.98096; 


.21132 


.97742 


.22835 


.97358 


.24531 


.96945 


48 


13 


.17737 


.98414 


.19452 


.98C90! 


.21161 


.97735 


.22863 


.97351 


.24559 


.96937 


47 


14 


.17766 


.98409 


.19481 


.98084 


.21189 


.97729 


.22892 


.97345 


.24587 


.96930 


46 


15 


.17794 


.98404 


.19509 


.98079 


.21218 


.97723 


.22920 


.97338 


.24615 


.96923 


45 


16 


.17823 


.98399 


.19538 


.98073! 


.21246 


.97717 


.22948 


.97331 


.24644 


.96916 


44 


17 


.17852 


.98394 


.19566 


. 98067 ' 


.21275 


.97711 


.22977 


.97325 


.24672 


.96909 


43 


18 


.17880 


.98389 


.19595 


.98061 j 


.21303 


.97705 


.23005 


.97318 


.24700 


.96902 


42 


19 


.17909 


.98383 


.19623 


.98056 


.21331 


.97698 


.23033 


.97311 


.24728 


.96894 


41 


20 


.17937 


.98378 


.19652 


.98050 


.21360 


.97692 


.23062 


.97304 


.24756 


.96887 


40 


21 


.17966 


.98373 


.19680 


. 98044 ' 


.21388 


.97686 


.23090 


.97298 


.24784 


.96880 


39 


22 


.17995 


.98368 


.19709 


.98039 


.21417 


.97680 


.23118 


.97291 


.24813 


.96873 


38 


23 


.18023 


.98362 


.19737 


.93033; 


.21445 


.97673 


.23146 


.97284 


.24841 


.96866 


37 


24 


.18052 


.98357 


.19766 


.98027 


.21474 


.97667 


.23175 


.97278 


.24869 


.96858 


36 


25 


.18081 


.98352 


.19794 


.98021 


.21502 


.97661! 


.23203 


.97271 


.24897 


.96851 


35 


26 


.18109 


.98347 


.19823 


.93016 


.21530 


.97655 


.23231 


.97264 


.24925 


.96844 


34 


27 


.18138 


.98341 


.19851 


.98010' 


.21559 


.97643 


.23260 


.97257 


.24954 


.96837 


33 


28 


.18166 


.98336 


.19880 


.98004; 


.21587 


.976421 


.23288 


.97251 


.249S2 


.96829 


32 


29 


.18195 


.98331 


.19908 


.97993 


.21616 


.97636 


.23316 


.97244 


.25010 


.96822 


31 


30 


.18224 


.98325 


.19937 


.97992: 


.21644 


.97630J 


.23345 


.97237 


.25038 


.96815 


30 


31 


.18252 


.98320 


.19965 


.97987 


.21672 


. 97623 ■ 


.23373 


.97230 


.25066 


.96807 


29 


32 


.18281 


.98315 


.19994 


.97981 


.21701 


.97617 


.23401 


.97223 


.25094 


.96800 


28 


33 


.18309 


.98310 


.20022 


.97975 


.21729 


.97611! 


.23429 


.97217 


.25122 


.96793 


27 


34 


.18338 


.93304 


.20051 


.97969 


.21758 


.976041 


.23458 


.97210 


.25151 


.96786 


26 


35 


.18367 


.93299 


.20079 


.97963 


.21786 


.97598| 


.23486 


.97203 


.25179 


.96778 


25 


36 


.18395 


.93294 


.23108 


.97958 


.21814 


.97592 


.23514 


.97196 


.25207 


.96771 


24 


37 


.18424 


.98288 


.20136 


.97952 


.21843 


.97585 


.23542 


.97189 


.25235 


.96764 


23 


38 


.18452 


.98283 


.20165 


.97946 


.21871 


.97579 


.23571 


.97182 


.25263 


.96756 


22 


39 


.18481 


.98277 


.20193 


.97940; 


.21899 


.97573; 


.23599 


.97176 


.25291 


.96749 


21 


40 


.18509 


.98272 


.20222 


.97934 


.21928 


.97566 


.23627 


.97169 


.25320 


.96742 


20 


41 


.18538 


.98267 


.20250 


.97928 


.21956 


.97560 


.23656 


.97162 


.25348 


.96734 


19 


42 


.18567 


.98261 


.20279 


. 97922 j 


.21985 


.97553 


.23684 


.97155 


.25376 


.96727 


18 


43 


.18595 


.98256, 


1 .20307 


.979161 


.22013 


.97547 


.23712 


.97148 


.25404 


.96719 


17 


44 


.18624 


.98250 


| .20336 


.979101 


.22041 


.97541 


.23740 


.97141 


.25432 


.96712 


16 


45 


.18652 


.98245' 


.20364 


.97905! 


.22070 


.97534 


.23769 


.97134 


.25460 


.96705 


15 


46 


.18681 


.98240| 


! .20393 


.97899 


.22098 


.97528: 


.23797 


.97127 


.25488 


.96697 


14 


47 


.18710 


.98234! 


! .20421 


.97893 


.22126 


.97521' 


.23825 


.97120 


.25516 


.96690 


13 


48 


.18738 


.98229 


I .204.50 


.97887| 


.22155 


.97515! 


.23853 


.97113 


.25545 


.96682 


12 


49 


.18767 


.98223 


1 .20478 


.97881 


.22183 


.97508; 


.23882 


.97106 


.25573 


.96675 


11 


50 


.18795 


.98218 


.20507 


.97875 


.22212 


.97502 


.23910 


.97100 


.25601 


.96667 


10 


51 


.18824 


.98212 


.20535 


.97869 


.22240 


.97496 


.23938 


.97093 


.25629 


.96660 


9 


52 


.18852 


.98207 


.20563 


.97863 


.22268 


.97489 


.23966 


.97086 


.25657 


.96653 


8 


53 


.18881 


.98201 


.20592 


.97857 


.22297 


.97483! 


.23995 


.97079 


.25685 .96645 


7 


54 


.18910 


.98196 


i .20620 


.97851 


.22325 


.97476 


.24023 


.97072 


.25713 .96638 


6 


55 


.18938 


.98190 


1 .20649 


.97845 


.22353 


.974701 


.24051 


.97065 


.25741 .96630 


5 


56 


.18967 


.98185 


! .20677 


.97839 


.22382 


.97463 


.24079 


.97058 


.25769 .96623 


4 


57 


.18995 


.98179 


1 .20706 


.97833 


.22410 


.97457 


.24108 


.97051 


.25798 .96615 


3 


58 


.19024 


.98174 


.20734 


.97827 


.22438 


.97450 


.24136 


.97044 


.25826 .96608 


2 


59 


.19052 


.98168 


.20763 


.97821 


.22467 


.97444 


.24164 


.97037 


.25854 .96600 


1 


60 
/ 


.19081 
Cosin 


.98163 


.20791 
Cosin 


.97815 
Sine 


.22495 
Cosin 


.97437 
Sine 


.24192 
Cosin 


.97030 


.25882 .96593 
Cosin Sine 



/ 


Sine. 


Sine 


79° 


78° 


77° 


76° 


75° 



TABLES. 



6 5 I 



TABLE VI. — Continued. 
Natural Sines and Cosines. 



t 


20° 


21° 


22° 


23° 


24° 


/ 


Sine 


Cosin 


Sine 


Cosin 


Sine 


Cosin 1 


Sine 


Cosin 


Sine Cosin 1 


"0 


.34202 


.93969 


.35837 


.93358 


.37461 


7927181 


.39073 


.92050 


740674 791355 


60 


1 


.34229 


.93959 


.35864 


.93348^ 


.37488 


.92707) 


.39100! 


.92039 


.40700 .91343 59 


2 


.34257 


.93949 


.35891 


.93337) 


.37515 


.92697' 


.39127 


.92028 


.40727 .91331 


58 


3 


.34284 


.93939! 


.35918 


. 93327 


.37542 


. 92686 j 


.39153 


.92016 


.40753 


.91319 


5? 


4 


.34311 


.93929 


.35945 


.933161 


.37569 


.92675 


.39180 


.92005 


.40780 


.91307 56 


5 


.34339 


.93919] 


.35973 


.93306 


.37595' 


.92664 


.39207 


.91994 


.40806 


.91295 


55 


6 


.34366 


.93909 


.36000 


.93295 


.37622 


.92653 


.39234 


.91982! 


.40833 


.91283 


54 


7 


.34393 


.93899 


.36027 


.93285 


.37649 


.92642 


.39260 


.91971: 


.40860 


.91272 


53 


8 


.34421 


.93889 


.36054 


.93274 


.37676 


.92631 


.39287 


.91959 ! 


.40886 


.91260 


52 


9 


.34448 


.93879 


.36081 


.932641 


.37703 


.92620 


.39314 


.91948 


.40913 


.91248 


51 


10 


.34475 


.93869! 


.36108 


.93253 


.37730 


.92609 


.39341 


.91936 


.40939 


.91236 


50 


11 


.34503 


.93859 


.361 35 


.93243 


.37757 


.92598 


: .39367 


.91925 


.40966' 


.91224 


49 


12 


.34530 


.938491 


.36102 


.93232 


.37784 


.92587 


1 .39394 


.91914 


.40992 


.91212 


48 


13 


. 34557 


.93839! 


.36190 


.93222 


.37811 


.92576 


.39421 


.91902 ' 


.41019 


.91200 


47 


14 


.34584 


.93S29S 


.36217 


.93211 


.37838 


.92565 


! .39448 


.91891 | 


.41045 


.91188 


46 


15 


.34612 


.938191 


.36244 


.93201 


.37865 


.92554 


! .39474 


.91879| 


.41072 


.91176 


45 


16 


.34639 


.938091 


.36271 


.93190 


.37892 


.92543 


1 .39501 


.91868: 


.41098 


.91164 


44 


17 


.34666 


.93799! 


.36298 


.93180 


.37919 


.92532 


.39528 


.91856! 


.41125 


.91152 43 


18 


.34694 


.93789 


.36325 


.93169 


.37946 


.92521 


.39555 


.91845: 


.41151 


.91140 42 


19 


.34721 


.93779' 


.36352 


.93159 


.37973 


.92510 


.39581 


.91833; 


.41178 


.91128 


41 


20 


.34748 


.93769! 


.36379 


.93148 


.37999 


.92499 


! .39608 


.91822 


.41204 


.91116 


40 


21 


.34775 


.93759 


.36406 


.93137 


.38026 


.92488 


' .39635 


.91810 


.41231 


.91104 


£3 


22 


.34803 


.93748 


.36434 


.93127 


.38053 


.92477 


.39661 


.91799 


.41257 


.91092 £3 


23 


.34830 


.93738 


.36461 


.93116 


.36080 


.92466 


.39688 


.91787 


.41284 


.91080 37 


24 


.34857 


.93728 


.36488 


.93106 


.38107 


.92455 


.39715 


.91775 


.41310 


.91068 30 


25 


.34884 


.93718 


i .36515 


.93095 


.38134 


.92444 


.39741 


.91764 


.41337 


.91056 35 


26 


.34912 


.93708' 


.36542 


.93084 


i .38161 


. 92432 | 


.39768 


.91752 


.41363 


.91044 


34 


27 


.34939 


.93698 


1 .36569 


.93074 


1 .38188 


.92421 


.39795 


.91741 


.41390 


.91032 


33 


28 


.34966 


.93688 


, .36596 


.93063 


.38215 


.92410 


.39822 


.91729 


.41416 


.91020 


32 


23 


.34993 


.93677 


1 .36623 


.93052 


.::8241 


.92399 


.39848 


.91718 


.41443 


.91008 


31 


30 


.35021 


.93667 


| .36650 


.93042 


.38268 


.92388 


.39875 


.91706 


.41469 


.90996 


30 


31 


.35048 


.9365? 


1 .36677 


.93031 


.38295 


.92377 


.39902 


.91694 


.41496 


.90984 


29 


32 


.35075 


.93647 


.36704 


.93020 


.38322 


.92366 


.39928 


.91683 


.41522 


.90972 


28 


33 


.35102 


.93637! 


.36731 


.93010 


.38349 


.92355 


.39955 


.91671 


! .41549 


.90960 


27 


34 


.35130 


.93626 


.36758 


.92999 


.38376 


.92343 


.39982 


.91660 


.41575 


.90948 


26 


35 


.35157 


.93616 


.36785 


.92988 


.38403 


.92332 


j. 40008 


.91648 


! .41602 


.90936 


25 


36 


.35184 


.93606 


.36812 


.92978 


.38430 


.92321 


| .40035 


.91636 


.41628 


.90924 


24 


37 


.35211 


.93596 


.36839 


.92967 


.38456 


.92310 


j .40062 


.91625 


.41655 


.90911 


23 


38 


.35239 


.93585 


.36867 


.92956 


.38483 


.92299 


| .40088 


.91613 


.41681 


.90899 


22 


39 


.35266 


.93575 


.36894 


.92945 


1 .38510 


.92287 


1 .40115 


.91601 


i .41707 


.90887 


21 


40 


.35293 


.93565 


.36921 


.92935 


.38537 


.92276 


.40141 


.91590 


.41734 


.90875 


20 


41 


.35320 


. 93555 '' 


.36948 


.92924 


.38564 


.92265 


.40168 


.91578 


.41760 


.90863 


19 


42 


.35347 


.93544 


.36975 


.92913 


.38591 


.92254 


.40195 


.91566 


.41787 


.90851 


18 


43 


.35375 


.93534 


.37002 


.92902 


.38617 


.92243 


.40221 


.91555 


i .41813 


.90839 


17 


44 


.35402 


.93524 


.37029 


.92892 


.38644 


.92231 


.40248 


.91543 


! .41840 


.90826 


16 


45 


.35429 


.93514 


.37056 


.928Sli 


.38671 


.92220 


.40275 


.91531 


.41866 


.90814 


15 


46 


.35456 


.93503 


.37083 


.928T0| 


.38698 


.92209 


.40301 


.91519 


j .41892 


.90802 


14 


47 


.35484 


.93493 


.37110 


.92859 


.38725 


.92198 


.40328 


.91508 


1 .41919 


.90790 


13 


48 


.35511 


.93483 


.37137 


.92849 


1 ■ 88752 


.92186 


.40355 


.91496 


i .41945 


.90778 


12 


49 


.35538 


.93472 


.37164 


.92838! 


1 .38778 


.92175 


.40381 


.91484 


' .41972 


.90766 


11 


50 


.35565 


.93462 


.37191 


.92827 


j .38805 


.92164 


.40408 


.91472 


1 .41998 


.90753 


10 


51 


.35592 


.93452 


.37218 


.92816 


.38832 


.92152 


.40434 


.91461 


! .42024 


.90741 


9 


52 


.35619 


.93441 


.37245 


.92805 


.38859 


.92141 


.40461 .91449 


j .42051 


.90729 


8 


53 


.35647 


.93431 


.37272 


.92794 


.38886 


.92130 


j .40488 .91437 


.42077 


.90717 


7 


54 


.35674 


.9:3420 


.37299 


.92784 


.38912 


.92119 


1 .40514 .91425 


.42104 


.90704 


6 


55 


.35701 


.93410 


.37326 


.92773 


.38939 


.92107: 


j .40541 .914141 


.42130 


.90692 


5 


56 


.35728 


.93400 


.37353 


.92762 


1 .38966 


.92096 


.40567 .91402 


1 .42156 


.90680 


4 


57 


. 35755 


.93389 


.37380 


.92751 


.38993 


.92085 


.40594 .91390! 


! .42183 


.90668 


3 


58 


.35782 


.93379 


.37407 


.927401 


.39020 


.92073 


.40621 .91378! 


.42209 


.90655 


2 


59 


.35810 


.93368 


.37434 


.92729! 


.39046 


.92062 


.40647 .91366 


: .42235 


.90643 


1 


60 
/ 


.35837 
Cosin 


.93358 
Sine 


.37461 
Cosin 


.92718 


.39073 
Cosin 


.92050 
Sine | 


.40674 .91355 
Cosin Sine 


1 .42262 
Cosin 


.90631 
Sine 



/ 


Sine 


69° 


68° 


1 . 67° 


66° 


65° 



652 



SURVEYING. 



TABLE Ml.— Continued. 
Natural Sines and Cosines. 



"0 


15° 


16° 


17° 


18° 


19° 


/ 

60 


Sine 

725882 


Cosin 

.96593 


Sine 
.27564 


Cosin 

.96126 


Sine 

.29237 


Cosin 
.95630 


Sine 

.30902 


Cosin 


Sine 

732557 


Cosin 


.95106 


.94552 


1 


.25910 


.96585 


.27592 


.96118 


.29265 


.95622 


.30929 


.95097 


1 .32584 


.94542 


59 


2 


.25938 


.96578 


.27620 


.96110 


.29293 


.95613 


.30957 


.95088 


.32612 


.94533 


58 


3 


.25966 


.96570 


.27648 


.96102 


.29321 


.95605 


.30985 


.95079 


.32639 


.94523 


57 


4 


.25994 


.96562 


.27676 


.96094 


.29348 


.95596 


.31012 


.95070 


.32667 


.94514 


56 


5 


.26022 


.96555 


.27704 


.96086 


.29376 


.95588 


i .31040 


.95061 


.32694 


.94504 


55 


6 


.26050 


.96547 


.27731 


.96078 


.29404 


.95579 


.31068 


.95052 


.32722 


.94495 


54 • 


7 


.26079 


.96540 


.27759 


•96070 


.29432 


.95571 


.31095 


.95043 


.32749 


.94485 


53 


8 


.26107 


.96532 


.27787 


.96062 


.29460 


.95562 


.31123 


.95033 


.32777 


.94476 


52 


9 


.26135 


.96524 


.27815 


.96054 


.29487 


.95554 


.31151 


.95024 


! .32004 


.94466 


51 


10 


.26163 


.96517 


.27843 


.96046 


.29515 


.95545 


.31178 


.95015 


.32832 


.94457 


50 


11 


.26191 


.96509 


.27871 


.96037 


.29543 


.95536 


I .31206 


.95006 


' .32859 


.94447 


4k 


12 


.26219 


.96502 


.27899 


.96029 


.29571 


.95528 


! .31233 


.94997 


■ .32887 


.94438 


48 


13 


.26247 


.96494! 


.27927 


.96021 


.29599 


.95519 


! .31261 


.94988 


.32914 


.94428 


47 


14 


.26275 


.96486! 


.27955 


.96013 


.29626 


.95511 


1 .31289 


.94979 


.32942 


.94418 


46 


15 


.26303 


.96479' 


.27983 


.96005 


.29654 


.95502 


.31316 


.94970 


: .32969 


.94409 


45 


10 


.26331 


.96471! 


.28011 


.95997 


.29682 


.95493 


.31344 


.94961 


.32997 


.94399 


44 


17 


.26359 


.96463' 


.28039 


.95989 


.29710 


.95485 


.31372 


.94952 


.33024 


.94390 


43 


18 


.26387 


.96456 


.28067 


.95981 


.29737 


.95476 


.31399 


.94943 


.33051 


.94380 


42 


19 


.26415 


.96448 


.28095 


.95972 


.29765 


.95467 


.31427 


.94933 


! .33079 


.94370 


41 


20 


.26443 


.96440 


.28123 


.95964 


.29793 


.95459 


.31454 


.94924 


.33106 


.94361 


40 


21 


.26471 


.96433 


.28150 


.95956 


.29821 


.95450 


.31482 


.94915 


' .33134 


.94351 


39 


22 


.26500 


.96425 


.28178 


.95948 


.29849 


.95441 


.31510 


.94906 


.33161 


.94342 


38 


23 


.26528 


.96417 


.28206 


.95940 


.29876 


.95433 


.31537 


.94897 


! .33189 


.94332 


37 


24 


.26556 


.96410 


.28234 


.95931 


.29904 


.95424 


.31565 


.94888 


I .33216 


.94322 


36 


25 


.26584 


.96402 


.28262 


.95923 


.29932 


.95415 


.31593 


.94878 


! .33244 


.94313 


35 


26 


.26612 


.96394 


.28290 


.95915 


.29960 


.95407 


.31620 


.94869 


1 .33271 


.94303 


34 


27 


.26640 


.96386 


.28318 


.95907 


.29987 


.95398 


.31648 


.94860 


.33298 


.94293 


33 


28 


.26668 


.96379 


.28346 


.95898 


.30015 


.95389 


.31675 


.94851 


.33326 


.94284 


32 


29 


.26696 


.96371 


.28374 


.95890 


.30043 


.95380 


.31703 


.94842 


1 .33353 


.94274 


31 


30 


.26724 


.96363. 


.28402 


.95882 


.30071 


.95372 


.31730 


.94832 


.33381 


.94264 


30 


31 


.26752 


.96355 


.28429 


.95874 


.30098 


.95363 


.31758 


.94823 


1 .33408 


.94254 


29 


32 


.26780 


.963471 


.28457 


.95865 


.30120 


.95354 


.31786 


.94814 


.33436 


.94245 


28 


33 


.26808 


.96340; 


.28485 


.95857 


.30154 


.95345 


.31813 


.94805 


.33463 


.94235 


27 


34 


.26836 


.963321 


.28513 


.95849 


.30182 


.95337 


.31841 


.94795 


.83490 


.94225 


26 


35 


.26864 


.96324 


.28541 


.95841 


.30209 


.95328 


.31868 


.94786 


.33518 


.94215 


25 


36 


.26892 


.96316 


.28569 


.95832 


.30237 


.95319 


.31896 


.94777 


.33545 


.94206 


24 


37 


.26920 


.96308 


.28597 


.95824 


.30265 


.95310 


.31923 


.94768 


.33573 


.94196 


23 


38 


.26948 


.96301; 


.28625 


.95816 


.30292 


.95301 


.31951 


.94758 


.33600 


.94186 


22 


39 


.26976 


.96293' 


.28652 


.95807 


.30320 


.95293 


.31979 


.94749 


.33627 


.94176 


21 


40 


.27004 


.96285; 


.28680 


.95799 


.30348 


.95284 


.32006 


.94740 


.33655 


.94167 


20 


41 


.27032 


.9627?' 


.28708 


.95791 


.30376 


.95275 


.32034 


.94730 


.33682 


.94157 


19 


42 


.270G0 


.962691 


.28736 


.95782 


.30403 


.95266 


.32061 


.94721 


.33710 


.94147 


18 


43 


.27088 


. 96261 | 


.28764 


.95774 


.30431 


.95257 


.32089 


.94712 


.38737 


.94137 


17 


44 


.27116 


.962531 


.28792 


.95766 


.30459 


.95248 


.32116 


.94702 


.33764 


.94127 


16 


45 


.27144 


.96246 


.28820 


.95757 


.30486 


.95240 


.32144 


.94693 


.33792 


.94118 


15 


46 


.27172 


.96238 


28847 


.95749 


.30514 


.95231 


.32171 


.94684 


.33819 


.94108 


14 


47 


.27200 


.96230' 


128875 


.95740 


.30542 


.95222 


.32199 


.94674 


.33846 


.94098 


13 


48 


.27228 


.96222 


.28903 


.95732 


.30570 


.95213 


.32227 


.94665 


.33874 


.94088 


12 


49 


.27256 


.96214 


.28931 


.95724 


.30597 


.95204 


.32254 


.94656 


33901 


.94078 


11 


50 


.27284 


.96206 


.28959 


.95715 


.30625 


.95195 


.32282 


.94646 


,.33929 


.94068 


10 


51 


.27312 


.96198 


.28987 


.95707 


.30653 


.95186 


.32309 


.94637 


.33956 


.94058 


9 


52 


.27340 


.961901 


.29015 


.95698 


.30680 


.95177 


.32337 


.94627 


! .33983 


.94049 


8 


53 


.27368 


.961821 


.29042 


.95690 


.30708 


.95168 


.32364 


.94618 


.34011 


.94039 


7 


54 


.27396 


.96174 


.29070 


.95681 


.30736 


.95159! 


.32392 


.94609 


1 .34038 


.94029 


6 


55 


.27424 


.96166; 


.29098 


.95673 


.30763 


.95150; 


.32419 


.94599 


! .34065 


.94019 


5 


56 


.27452 


. 96158 ! 


.29126 


.95664 


.30791 


.95142; 


.32447 


.94590 


! .34093 


.94009 


4 


57 


.27480 


.96150: 


.29154 


.95656 


.30819 


.95133 


.32474 


.93580 


.34120 


.93999 


3 


58 


.27508 


.96142 


.29182 


.95647 


.30846 


.95124 


.32502 


.94571 


.34147 


.93989 


2 


59 


.27536 


.96134 


.29209 


.95639 


.30874 


.95115 


.32529 


.94561 


.34175 


.93979 


1 


60 
/ 


.27564 
Cosin 


.96126 
Sine 


.29237 
Cosin 


.95630 
Sine 


.30902 
Cosin 


.95106 
Sine 


.32557 
Cosin 


.94552 
Sine 


.34202 
Cosin 


.93969 



t 


Sine 


74° 


73° 


72° 


71° 


70° 



452 



TABLES. 



653 



TABLE VI.— Continued. 
Natural Sines and Cosines. 



/ 

~0 


25° 


26° 


27° 


|. 28° 


29° 


/ 

60 


Sine 


Cosin 
.90631 


Sine 
743837 


Cosin 

.89879 


Sine 
.45399 


Cosin 

.89101 


Sine 
1 .46947 


Cosin 


Sine 


Cosin 

.87462 


.42262 


.882951 


.48481 


1 


.42288 


.90618 


.43863 


.89867 


.45425 


.89087 


.46973 


.88281 


.48506 


.87448 


59 


2 


.42315 


.90606 


.43889 


.89854 


.45451 


.89074 


.46999 


.88267 


.48532 


.87434 


58 


3 


.42341 


.90594 


.43916 


.89841 


.45477 


.89061 


.47024 


.88254 


.48557 


.87420 


57 


4 


.42367 


.90582 


.43942 


.89828 


.45503 


.89048 


.47050 


.88240 


.48583 


.87406 


56 


5 


.42394 


.90569 


.43968 


.89816 


.45529 


.89035 


.47076 


.88226 


.48608 


.87391 


55 


6 


.42420 


.90557 


.43994 


.89803 


.45554 


.89021 


.47101 


.88213 i 


.48634 


.87377 


54 


7 


.42446 


.90545 


.44020 


.89790 


.45580 


.89008 


.47127 


.88199 : 


.48659 


.87363 


53 


8 


.42473 


.90532 


.44046 


.89777 


.45606 


.88995 


.47153 


.88185 


.48684 


.87349 


52 


9 


.42499 


.90520 


.44072 


.89764 


.45632 


.88981 


.47178 


.88172 | 


.48710 


.87335 


51 


10 


.42525 


.90507 


.44098 


.89752 


.45658 


.88968 


.47204 


.88158 


.48735 


.87321 


50 


11 


.42552 


.90495 


.44124 


.89739 


.45684 


.88955 


.47229 


.88144 


.48761 


.87306 


4? 


12 


.42578 


.90483 


.44151 


.89726 


.45710 


.88942 


.47255 


.88130 


.48786 


.87292 


48 | 


13 


.42604 


.90470 


.44177 


.89713 


.45736 


.88928 


.47281 


.88117J 


.48811 


.87278 


47 


14 


.42631 


.90458 


.44203 


.89700 


.45762 


.88915 


.47306 


.88103 


.48837 


.87264 


46 


15 


.42657 


.90446 


.44229 


.89687 


.45787 


.88902 


.47332 


.88089 


.48862 


.87250 


45 


16 


.42683 


.90433 


.44255 


.89674 


.45813 


.88888 


.47358 


.88075 


.48888 


.87235 


44 


17 


.42709 


.90421 


.44281 


.89662 


.45839 


.88875 


.47383 


.88062 


.48913 


.87221 


43 


18 


.42736 


.90408 


.44307 


.89649 


.45865 


.88862 


.47409 


.88048 


.48938 


.87207 


42 


19 


.42762 


.90396 


.44333 


.89636 


.45891 


.88848 


.47434 


.88034 


.48964 


.87193 


41 


20 


.42788 


.90383 


.44359 


.89623 


.45917 


.88835 


.47460 


.88020 


.48989 


.87178 


40 


21 


.42815 


.90371 


.44385 


.89610 


.45942 


.88822 


.47486 


.88006 


.49014 


.87164 


39 


22 


.42841 


.90353 


.44411 


.89597 


.45968 


.88808 


.47511 


.87993 


.49040 


.87150 


38 


23 


.42867 


.90346 


.44437 


.89584; 


.45994 


.88795 


.47537 


.87979 


.49065 


.87136 


37 


24 


.42894 


.903:34 


.44464 


.89571 


.46020 


.88782 


.47562 


.87965 


.49090 


.87121 


36 


25 


.42920 


.90321 


.44490 


.89558 


.46046 


.88768 


.47588 


.87951 


.49116 


.87107 


35 


26 


.42946 


.90309 


.44516 


.89545 


.4G072 


.88755 


.47614 


.87937 


.49141 


.87093 


34 


27 


.42972 


.90296 


.44542 


.89532 


.4G097 


.88741 


.47639 


.87923 


.49166 


.87079 


33 


28 


.42999 


.90284 


.44568 


.89519 


.46123 


.88728 


.47665 


.87909 


.49192 


.87064 


32 


29 


.43025 


. 9i"!271 


.44594 


.89506 


.46149 


.88715 


.47690 


.87896 


.49217 


.87050 


31 


30 


.43051 


.90259 


.44620 


.89493 


.46175 


.88701 


.47716 


.87882 


.49242 


.87036 


30 


31 


.43077 


.90246 


.44646 


.89480 


.46201 


.88688 


.47741 


.87868 


.49268 


.87021 


29 


32 


.43104 


.90233 


.44672 


.89467 


.46226 


.88674 


.47767 


.87854 


.49293 


.87007 


28 


33 


.43130 


.90221 


.44698 


.89454 


.46252 


.88661 


.47793 


.87840 


.49318 


.86993 


27 


34 


.43156 


.90208 


.44724 


.89441 


.46278 


.88647 


.47818 


.87826 


.49344 


.86978 


26 


35 


.43182 


.90196 


.44750 


.89428 


.46304 


.88634 


.47844 


.87812 


.49369 


.86964 


25 


36 


.43209 


.90183 


.44776 


.89415 


.46330 


.88620 


.47869 


.87798 


.49394 


.86949 


24 


37 


.43235 


.90171 


.44802 


.89402 


.46355 


.88607 


.47895 


.87784 


.49419 


.86935 


23 


38 


.43261 


.90158 


.44828 


.89389 


.46381 


.88593 


.47920 


.87770 


.49445 


.86921 


22 


39 


.43287 


.90146 


.44854 


.89376 


.46407 


.88580 


.47946 


.87756 


.49470 


.86906 


21 


40 


.43313 


.90133 


.44880 


.89363 


.46433 


.88566 


.47971 


.87743 


.49495 


.86892 


20 


41 


.43340 


.90120 


.44906 


.89350 


.46458 


.88553 


.47997 


.87729 


.49521 


.86878 


19 


42 


.43366 


.90108 


.44932 


.89337 


.464S4 


.88539 


.48022 


.87715 


.49546 


.86863 


18 


43 


.43392 


.90095 


.44958 


.89324 


.46510 


.88526 


.48048 


.87701 


.49571 


.86849 


17 


44 


.43418 


.90082 


.44984 


.89311 


.46536 


.88512 


.48073 


.876S7 


.49596 


.86834 


16 


45 


.43445 


.90070 


.45010 


.89298 


.46561 


.88499 


.48099 


.87673 


.49622 


.86820 


15 


46 


.43471 


.90057 


.45036 


.89285 


.46587 


.88485 


.48124 


.87659 


.49647 


.86805 


14 


47 


.43497 


.90045! 


.45062 


.89272 


.46613 


.88472 


.48150 


.87645 


.49672 


.86791 


13 


48 


.43523 


.90032 


.45088 


.89259 


! .46639 


.88458 


.48175 


.87631 


! .49697 


.86777 


12 


49 


.43549 


.90019 


.45114 


.89245 


I .46664 


.88445 


.48201 


.87617 


| .49723 


.86762 


11 


50 


.43575 


. 90007 | 


.45140 


.89232 


.46690 


.88431 


.48226 


.87603 


.49748 


.86748 


10 


51 


.43602 


.89994 


.45166 


.89219 


.46716 


.88417 


.48252 


.87589 


.49773 


.86733 


9 


52 


.43628 


.89981 


.45192 


.89206 


1 .46742 


.88404 


.48277 


.87575 


.49798 


.86719 


8 


53 


.43654 


.89968 


.45218 


.89193 


! .46767 


.88390 


.48303 


.87561 


1 .49824 


.86704 


1 


54 


.43680 


.89956! 


.45243 


.89180 


.46793 


.88377 


.48328 


.87546 


.49849 


.86690 


6 


55 


.43706 


.89943, 


.45269 


.89167 


.46819 


.88363 


.48354 


.87532 


.49874 


.86675 


5 


56 


.43733 


.89930J 


.45295 


.89153 


.46844 


.88349 


.48379 


.87518 


.49899 


.86661 


4 


57 


.43759 


.89918 


.45321 


.89140 


.46870 


.88336 


.48405 


.87504 


.49924 


.86646 


3 


58 


.43785 


.89905 


.45347 


.89127 


.46896 


.88322 


.48430 


.87490 


.49950 


.86632 


2 


59 


.43811 


.89892 


.45373 


.89114 


.46921 


.88308 


.48456 


.87476 


.49975 


.86617 


1 


60 
t 


.43837 
Cosin 


.89879 
Sine 1 


.45399 
Cosin 


.89101 
Sine 


' .46947 
Cosin 


.88295 
Sine 


.48481 
1 Cosin 


.87462 
Sine 


.50000 
Cosin 


.86603 


_0 
1 

\ 


Sine 


64° 


63° 


62° 


61° 


60° 



654 



SURVEYING. 



TABLE VI. — Continued. 
Natural Sines and Cosines. 



/ 

~0 


30- 


31° 


32° 


33° 


34° 


/ 

60 


Sine 


Cosin 

.86603 


Sine 
751504 


Cosin 
.85717 


Sine 


Cosin 

.84S05 


Sine 

.54464 


Cosin 


Sine 
.55919 


Cosin 


.50000 


.52992 


.83867 


.82904 


1 


.50025 


.86588 


.51529 


.85702 


.53017 


.84789 


.54488 


.83851 


.55943 


.82887 


59 


2 


.50050 


.86573 


.51554 


.85687 


.53041 


.84774 


.54513 


.83835 


.55968 


.82871 


58 


3 


.50076 


.86559 


.51579 


.85672 


.53066 


.84759 


.54537 


.83819 


.55992 


.82855 


£7 


4 


.50101 


.86544 


.51604 


.85657 


.53091 


.84743! 


.54561 


.83804 


.56016 


.82839 


58 


5 


.50126 


.86530 


.51628 


.85642 


.53115 


.84728 


.54586 


.83788 


.56040 


.82822; 


55 


6 


.50151 


.86515 


.51653 


.85627 


.53140 


.84712 


.54610 


.83772 


.56064 


.82806 


54 


7 


.50176 


.86501 


.51678 


.85612 


.53164 


.84697 


.54635! 


.83756 


.56088 


.82790 


53 


8 


.50201 


.86486 


.51703 


.85597 


.531 89 


.84681 


. 54659 i 


.83740 


.56112 


.82773 


52 


9 


.50227 


.86471 


.51728 


.85582 


.53214 


.84666 


.54683| 


.83724 


.56136 


.82757 


5i 


10 


.50252 


.86457 


.51753 


.85567 


.53238 


.84650 


.54708 


.83708 


.56160 


.82741 


50 


11 


.50277 


.86442 


.51778 


.85551 


.53263 


. 84635 j 


.54732 


.83692 


.56184 


.82724 


49 


12 


.50302 


.86427 


.51803 


.85536 


.53288 


.84619 


.54756 


.83676 


.58208 


.82708 


48 


13 


.50327 


.86413 


.51828 


.85521 


.53312 


.84604: 


.54781 


.83660 


.56232 


.82692 


47 


14 


.50352 


.86398 


.51852 


.85506 


.53337 


.84588 


.54805 


.83645 


.56256 


.82675 


46 


15 


.50377 


.86384 


.51877 


.85491 


.53361 


.84573 


.54829 


.83629 


.56280 


.82659 


45 


16 


.50403 


.86369 


.51902 


.85476 


.53386 


. 84557 i 


.54854 


.83613 


.56305 


.82643 


44 


17 


.50428 


.86354 


.51927 


.85461 


.53411 


.84542 


.54878 


.83597 


.56329 


.82626 


43 


18 


.50453 


.86340 


.51952 


.85446 


.53435 


.84526 


.54902 


.83581 


.56353 


.82610 


42 


19 


.50478 


.86325 


.51977 


.85431 


.53460 


.84511 


.54927 


.83565 


.56377 


.82593 


41 


20 


.50503 


.86310 


.52002 


.85416 


.53484 


.84495i 


.54951 


.83549 


.56401 


.82577 


40 


21 


.50528 


.86295 


.52026 


.85401 


.53509 


. 84480 ' 


.54975 


.83533 ! 


.56425 


.82561 


39 


22 


.50553 


.86281 


.52051 


.85385 


.53534 


.84464 


.54999 


.83517 


.56449 


.82544 


38 


23 


.50578 


.86266 


.52076 


.85370 


.53558 


.84448 


.55024 


.83501! 


.56473 


.82528 37 


24 


.50603 


.86251 


.52101 


.85355 


.53583 


.84433 


.55048 


.834851 


.56497 


.82511 


36 


25 


.50628 


.86237 


.52126 


.85340 


.53607 


.84417: 


.55072 


.83469 


.56521 


.82495 


35 


26 


.50654 


.86222 


.52151 


.85325 


.53632 


.84402 


.55097 


.83453 


.56545 


.82478 34 


27 


.50679 


.86207 


.52175 


.85310 


.53656 


.84386 


.55121 


.83437 j 


.56569 


.82462 33 


28 


.50704 


.86192 


.52200 


.85294 


.53681 


.84370 


.55145 


.83421! 


.56593 


.82446 32 


29 


.50729 


.86178 


.52225 


.85279 


.53705 


.84355 


.55169 


.83405 


.56617 


.82429 31 


30 


.50754 


.86163 


.52250 


.85264 


.53730 


.84339 


.55194 


.83389 


.56641 


.82413 j 30 


31 


.50779 


.86148 


.52275 


.85249' 


.53754 


.84334 ^ 


.55218 


.83373 


.56665 


.82396 


29 


32 


.50804 


.86133 


.52299 


.85234 


.53779 


.84308 


.55242 


.83356 


.56689 


.82380 


28 


33 


.50829 


.86119 


.52324 


.85218 


.53804 


.84292 


.E5266 


.83340: 


.56713 


.82303 


27 


34 


.50854 


.86104 


.52349 


.85203 


.53828 


.842771 


.55291 


.83324 


.56736 


.82347 


26 


35 


.50879 


.86089 


.52374 


.85188 


.53853 


.84261 i 


,55315 


.83308 


.56760 


.82330 


25 


36 


.50904 


.86074 


.52399 


.85173 


.53877 


.84245 


.55339 


.83292 


.56784 


.82314' 24 


37 


.50929 


.86059 


.52423 


.85157 


.53902 


.84230 


.55363 


.83276 


.56808 


.82297- 23 


38 


.50954 


.86045 


.52448 


.85142 


.53926 


.84214 


.55383 


.83260 


.56832 


.82281 22 


39 


.50979 


.86030 


.52473 


.85127 


.53951 


.84198 


.55412 


.83244 


.56856 


.82264 21 


40 


.51004 


.86015 


.52498 


.85112 


.53975 


.84182, 


.55436 


.83228 


.56880 


.82248 


20 


41 


.51029 


.86000 


.52522 


.85096! 


.54000 


.84167 


.55460 


.83212 


.56904 


.82231 


19 


42 


.51054 


.85985 


.52547 


.85081 


.54024 


.84151 


.55484 


.83195 


.56928 


.82214 


18 


43 


.51079 


.85970 


.52572 


.83066 


.54049 


.84135 


.55509 


.83179 


.56952 


.82198 


17 


44 


.51104 


.85956 


.52597 


.85051 


.54073 


.84120 


.55533 


.83163 


.56976 


.82181 


16 


45 


.51129 


.85941 


.52621 


.85035 


.54097 


.84104 


.55557 


.83147 


.57000 


.82165 


15 


46 


.51154 


.85926 


.52646 


.85020 


.54122 


.84088 


.55581 


.83131 


.57024 


.82148 


14 


47 


.51179 


.85911 


.52671 


.85005 


.54146 


.84072 


.55605 


.83115 


.57047 


.82132 


13 


48 


.51204 


.85896 


.52696 


.849S9 


.54171 


.84057 


.55630 


.83098 


I .57071 


.82115 


12 


49 


.51229 


.85881 


.52720 


.84974 


.54195 


.84041 


.55654 


.83082 


j .57095 


.82098 


11 


50 


.51254 


j. 85866 


.52745 


.84959 


.54220 


,84025 


. 55678 


.83066 


.57119 


.82082 


10 


51 


.51279 


.85851 


.52770 


.84943 


.54244 


.84009 


.55702 


.83050 


.57143 


.82065 


9 


52 


.51304 


1.83836 


.52794 


.84928 


.54269 


.83994 


.55726 


.83034 


.57167 


.82048 


8 


53 


.51329 


1. 85821 


.52819 


.84913 


.54293 


.83978 


.55750 


.83017 


.57191 


.82032 


7 


54 


.51354 


1.83806 


.52844 


.84897 


.54317 


.83962 


.55775 


.83001 


.57215 


.82015 


6 


55 


.51379 


.85792 


.52869 


.84882 


.54342 


.83946 


.55799 


.82985 


.57238 


.81999 


5 


56 


.51404 


1.85777 


.52893 


.84866 


.54366 


.83930 


.55823 


.82969 


.57262 


.81982 


4 


57 


.51429 


.85762 


.52918 


.84851 


.54391 


.83915 


.53847 


.82953 


.57286 


.81965 


3 


58 


.51454 


.85747 


.52943 


.84836 


.54415 


.83899 


1 .55871 


.82936 


.57310 


.81949 


2 


59 


.51479 


1.85732 


.52967 


.84820 


.54440 


.83883 


.55895 


.82920 


.57334 


.81932 


1 


60 


.51504 


|. 85717 


.52992 


.84805 


.54464 


1.83867 


.55919 


.82904 


.57358 


.81915 


1 


/ 


Cosin j Sine 


Cosin 


Sine 


Cosin 1 Sine 


Cosin 


Sine 


Cosin 


Sine 


• 


59° 


58° 


57° 


56° 


55° 



TABLES. 



655 



TABLE VI. — Continued. 
Natural Sines and Cosines. 



"0 


35° 


36° 


37° 


38° 


39° 


/ 

60 


Sine 


Cosin 
.81915 


Sine 

.58779 


Cosin 

.80902 


Sine 

.60182 


Cosin 

.79864 


Sine 
.61566 


Cosin 

778801 


Sine 

.62932 


Cosin 


.57358 


.77715 


1 


.57381 


.81899 


.58802 


.80885 


.60205 


.79846 


.61589 


.78783 


.62955 


.77696 


59 


2 


.57405 


.81882 


.58826 


.80867] 


.60228 


.79829 


.61612 


.78765 


.62977 


.77678 


58 


3 


.57429 


.81865 


.58849 


.808501 


.60251 


.79811 


.61635 


.78747 


.63000 


.77660 


57 


4 


.57453 


.81848 


.58873 


.80833! 


.60274 


.79793 


.61058 


.78729 


.63022 


.77641 


56 


5 


.57477 


. 81832 i 


.58896 


.80816; 


.60298 


.79776 


.61681 


.78711 


.63045 


.77623 


55 


6 


.57501 


.81815 ' 


.58920 


.80799 


.60321 


.79758 


.61704 


.78694 


.63068 


.77605 


54 


7 


.57524 


.817981 


.58943 


.80782 


.60344 


.79741 


.61726 


.78676 


.63090 


.77586 


53 


8 


.57548 


.81782! 


.5S907 


.80765 


.60367 


.79723 


.61749 


.78658 


.63113 


.77568 


52 


9 


.57572 


.81705 


.5S990 


.80748 


.60390 


.79706 


.61772 


.78640 


.63135 


.77550 


51 


10 


.57596 


.81748! 


.59014 


.80730 


.60414 


.79688 


.61795 


.78622 


.63158 


.77531 


50 


11 


.57619 


.81731 


.59037 


.80713 


.60437 


.79671 


.61818 


.78604 


.63180 


.77513 


40 


12 


.57643 


.81714 


.59001 


.80696 


.60460 


.79653 


.61841 


.78586 


.63203 


.77494 


48 


13 


.57667 


.8169S 


.59084 


.80679 


.60483 


.79635 


.61864 


.78508 


.63225 


.77476 


47 


14 


.57691 


.81681 


.59108 


.806621 


.60506 


.79618 


.61887 


.78550 


.63248 


.77458 


46 


15 


.57715 


.81664 


.59131 


.806441 


.60529 


.79600 


.61909 


.78532 


.63271 


.77439 


45 


16 


.57738 


.81647 


.59154 


.80627! 


.60553 


.79583 


.61932 


.78514 


.63293 


.77421 


44 


17 


.57762 


.81631 


.59178 


.80610 


.60576 


.79505 


.61955 


.78496 


.63310 


.77402 


43 


18 


.57786 


.81014 


.59201 


.80593! 


.60599 


.79547 


.61978 


.78478 


.63338 


.77384 


42 


19 


.57810 


.81597 


.59225 


.80576 


.60622 


.79530 


.62001 


.78460 


.63301 


.77366 


41 


20 


.57833 


.81580 


.59248 


.80558, 


.60645 


.79512 


.62024 


.78442 


.63383 


.77347 


40 


21 


.57857 


.81563 


.59272 


.80541 1 


.60668 


.79494 


.62046 


.78424 


.63406 


.77329 


39 


22 


.57881 


.815461 


.59295 


.80524 


.60691 


.79477 


.62069 


.78405 


.63428 


.77310 


38 


23 


.57904 


.81530 


.59318 


.80507 


.60714 


.79459 


.62092 


.78387! 


.63451 


.77292 


37 


24 


.57928 


.81513 


.59342 


.80489 


.60738 


.79441 


.62115 


.78369 


.63473 


.77273 


36 


25 


.57952 


.81496 


.59365 


.80472 


.60761 


.79424 


.62138 


.78351 


.63496 


.77255 


35 


23 


. 57976 


.81479 


.59389 


.80455 


.60784 


.79406 


.62160 


.78333 


.63518 


.77236 


34 


27 


.57999 


.81462J 


.59412 


.80438 


.60807 


.79388 


.62183 


.78315 


.63540 


.77218 


33 


28 


.58023 


.814451 


.59436 


.80420 


.60830 


.79371 


.62206 


.78297. 


.63563 


.77199 


32 


20 


.58047 


.81428! 


.59459 


.80403 


60853 


.79353 


.62229 


.78279' 


.63585 


.77181 


31 


30 


.58070 


.81412 


.59482 


.80386 


.60876 


.79335 


.62251 


.78261 


.63608 


.77162 


30 


31 


.58094 


.81395! 


.59506 


.80308 ' 


.60899 


.79318 


.62274 


.78243 


.63630 


.77144 


29 


32 


.58118 


.81378^ 


.59529 


.80351 


.60922 


.79300 


.62297 


.78225 


.63653 


.77125 


28 


33 


.58141 


.81361! 


.59552 


.80334 


.60945 


.79282 


.62320 


.78206 


.63675 


.77107 


27 


34 


.58165 


.81344! 


.59576 


.80316 


.60968 


.79264 


.62342 


.78188 


.63698 


.77088 


26 


35 


.58189 


.81327 


.59599 


.80299; 


.60991 


.79247 


.62365 


.78170 


.63720 


.77070 


25 


36 


.58212 


.81310 


.59022 


.80282 


.61015 


.79229 


.62388 


.78152! 


.63742 


.77051 


24 


37 


.58236 


.81293 


.59040 


.80264! 


.61038 


.79211 


.62411 


.78134 


.63765 


.77033 


23 


38 


.58260 


.81276! 


.59009 


.80247, 


.61061 


.79193 


.62433 


.78116: 


.63787 


.77014 


22 


39 


.58283 


.81259i 


.59693 


.80230; 


.61084 


.79176 


.62456 


.78098 


.63810 


.70990 


21 


40 


.58307 


.81242 


.59716 


.80212 


.61107 


.79158 


.62479 


.78079] 


.63832 


.76977 


20 


41 


.58330 


.81225 


.59739 


.80195 


.61130 


.79140 


.62502 


.78061 1 


.63854 


.76959 


19 


42 


.58354 


.81208 


.59763 


.80178 


.61153 


.79122 


.62524 


.78043 


.63877 


.76940 


18 


43 


.58378 


.81191 ! 


.59786 


.80160 


.61176 


.79105 


.62547 


.78025 


.63899 


.76921 


17 


44 


.58401 


.81174 


.59809 


.80143 


.61199 


.79087 


.62570 


.78007 


.63922 


.76903 


16 


45 


.58425 


.81157! 


.59832 


.80125 ] 


.61222 


.79069 


.62592 


.77988 


.63944 


.70884 


15 


46 


.58449 


.81140! 


.59856 


.80108 


.61245 


.79051 


.62615 


.77970 


.63966 


.76866 


14 


47 


.58472 


.81123! 


.59879 


.80091 1 


.61268 


.79033 


.62638 


.77952 


.63989 


.76847 


13 


48 


.58496 


.81106! 


.59902 


.80073! 


.61291 


.79016 


.62660 


.77934 


.64011 


.76828 


12 


49 


.58519 


.81089 


.59926 


.80056 


.61314 


.78998 


1 .62683 


.77916 


.04033 


.76810 


11 


50 


.58543 


.81072 


.59949 


.80038 


.61337 


.78980 


.62706 


.77897! 


.64056 


.76791 


10 


51 


.58567 


.81055 ' 


.59972 


. 80021 ; 


.61360 


.78962 


.62728 


.77879 


.64078 


.76772 


9 


52 


.58590 


.81038! 


.59995 


80003! 


.61383 


.78944 


.62751 


.77861 


.64100 


.76754 


8 


53 


.58614 


.81021 


.60019 


.79986' 


.61406 


.78926 


.62774 


.77843 


.64123 


.76735 


7 


54 


.58637 


.81004- 


.60042 


.79968 


.61429 


.78908 


.62796 


.77824; 


.64145 


.76717 


6 


55 


.58661 


.80987! 


.60065 


.79951 


.61451 


.78891 


.62819 


.77806 


.64167 


.76698 


5 


56 


.58684 


.80970! 


.60089 


.79934 


.61474 


.78873 


.62842 


.77788 


.64190 


.76679 


4 


57 


.58708 


.80953! 


.60112 


.79916 


.61497 


.78855 


.62864 


.77769 


.64212 


.76661 


3 


58 


.58731 


.80936 


.60136 


.79899 


.61520 


.78837 


.62887 


.77751 


.64234 


.76642 


2 


59 


.58755 


.80919 


.60158 


.79881 


.61543 


.78819 


.62909 


.77733 


.64256 .76623 


1 


60 

!' 


.58779 
Cosin 


.80902 
Sine 


.60182 
Cosin 


.79864 
Sine I 


.61566 
Cosin 


.78801 


.62932 
Cosin 


.77715 


.64279 .76604 
Cosin Sine 



/ 


Sine 


Sine 


54° 


53° 


52° 


51- 


50° 



656 



SURVEYING. 



TABLE VI.— Continued. 
Natural Sines and Cosines. 



r 
"0 


40° 


41° 


42° 


43° 


44° 


/ 

60 


Sine 
.61279 


Cosin 


Sine 


Cosin 


Sine 


Cosin 

.74314 


Sine 

768200 


! Cosin 


Sine 

.69466 


Cosin 
.71934 


.76604 


.65606 


.75471 


.66913 


.73135 


1 


.61301 


.76586 


.65628 


.75452 


.66935 


.74295 


.68221 


.73116 


.69487 


.71914 


59 


2 


.64323 


.76567 


.65650 


.75433 


.66956 


.74276 


.68242 


.73096 


.69508 


.71894 


58 


3 


.64346 


.76548 


.65672 


.75414 


.66978 


.74256 


.68264 


.73076 


.69529 


.71873 


57 


4 


.64368 


.76530 


.65694 


.75395 


.66999 


.74237 


.68285 


.73056 


.69549 


.71853 


56 


5 


.64390 


.76511 


.65716 


.75375 


/. 67021 


.74217 


.68306 


.73036 


.69570 


.71833 


55 


6 


.64412 


.76492 


.65738 


.75356 


.67043 


.74198 


.68327 


.73016 


.69591 


.71813 


54 


7 


.64435 


.76473 


.65759 


.75337 


.67064 


.74178 


.68349 


.72996 


.69612 


.71792 


53 


8 


.64457 


.76455 


.65781 


.75318 


.67086 


.74159 


.68370 


.72976 


.69633 


.71772 


52 


9 


.64479 


.76436 


.65803 


.75299 


.67107 


.74139 


.68391 


.72957 


.69654 


.71752 


51 


10 


.64501 


.76417 


.65825 


.75280 


.67129 


.74120 


.68412 


.72937 


.69675 


.71732 


50 


11 


.64524 


.76398 


.65847 


.75261 


.67151 


.74100 


.68434 


.72917 


.69696 


.71711 


49 


12 


.64546 


.76380 


.65869 


.75241 


.67172 


.74080 


.68455 


.72897 


.69717 


.71691 


48 


13 


.64568 


.76361 


.65891 


.75222 


.67194 


.74061 


.68476 


.72877 


.69737 


.71671 


47 


14 


.64590 


.76342 


.65913 


.75203 


.67215 


.74041 


.68497 


.72857 


.69758 


.71650 


<fc3 


15 


.64612 


.76323! 


.65935 


.75184 


.67237 


.74022 


.68518 


.72837 


.69779 


.71630 


45 


16 


.64635 


. 76304 j 


.65956 


.75165 


.67258 


.74002 


.68539 


.72817 


.69800 


.71610 


44 


17 


.64657 


.76236 


.65378 


.75146 


.67280 


.73983 


.68561 


.72797 


.69821 


.71590 


43 


18 


.64679 


.76267; 


.66000 


.75126 


.67301 


.73903 


.68582 


.72777 


.69842 


.71569 


42 


19 


.64701 


.7G243 : 


.63022 


.75107 


.67323 


.73944 


.68803 


.72757 


.69862 


.71549 


41 


20 


.64723 


. 76229 j 


.66044 


.75088 


.67344 


.73924 


.68624 


.72737 


.69883 


.71529 


40 


21 


.64746 


.76210 


.66066 


.75069 


.67366 


.73904 


.68645 


.72717 


.69904 


.71508 


39 


22 


.64768 


.761921 


.63038 


.75050! 


.67387 


.73885 


.68606 


.72697! 


.69925 


.71488 


38 


23 


.64790 


.76173! 


.68109 


.750301 


.67409 


.73865 


.68688 


.72677J 


.69946 


.71468 


37 


24 


.64812 


.76154 


.68131 


.75011 


.67430 


.73846 


.68709 


.72657 


.69966 


.71447 


36 


25 


.64834 


.76135 


.66153 


.74992; 


.67452 


.73823 


.68730 


.72637: 


.69987 


.71427 


35 


26 


.64856 


.761161 


.66175 


.74973 


.67473 


.73803 


.68751 


.72317! 


.70008 


.71407 


34 


27 


.64878 


.76097! 


.66197 


.74953 


.67495 


.73787 


.68772 


.72597' 


.70029 


.71386 


33 


28 


.64901 


.76078 


.68218 


.74934' 


.67516 


.73767 


.68793 


.72577! 


.70049 


.71366 


32 


29 


.64923 


.76059; 


.66240 


.74915 


.67538 


.73747 


.63814 


.72557 


.70070 


.71345 


31 


30 


.64945 


.76041 


.66262 


.74896 


.67559 


.73728 


.68835 


.72537 


.70091 


.71325 


30 


31 


.64967 


.76022 


.66284 


.74876' 


.67580 


.73708 


.68857 


.72517 


.70112 


.71305 


29 


32 ! .64933 


.76003 


.68306 


. 74857 j 


.67602 


.73683 


.63878 


.72497 


70132 


.71284 


28 


33 .65011 


.75934! 


.66327 


.74838: 


.67623 


.73669 


.63399 


.72477 


.70153 


.71264 


27 


34 .65033 


.75965! 


.66349 


.74818, 


.67645 


.73649 


.63920 


.72457 


.70174 


.71243 


26 


35 ! .65055 


.75946' 


.63371 


.74799, 


.67666 


.73623 


.68941 


.72437 


.70195 


.71223 


35 


36 .65077 


.75927 


.63393 


.747801 


.67688 


.73610 


.68902 


.72417 


.70215 


.71203 


24 


37 | .65100 


.75908 


.66414 


.74760! 


.67709 


.73590 


.68983 


.72397 


.70236 


.71182 


23 


38 .65122 


.75889 


.68436 


.74741 


.67730 


73570 


.69004 


.72377 


.70257 


.71162 


22 


39 I .65144 


.75870 


.66458 


.74722; 


.67752 


.73551 


.69025 


.72357 


.70277 


.71141 21 


40 


.65166 


.75851 j 


.66480 


.74703: 


.67773 


.73531 


.69046 


.72337 


.70298 


.71121 20 


41 


.65188 


.75832 


.66501 


.74683 


.67795 


.73511 


.69067 


.72317 


.70319 


.71100 19 


42 


.65210 


.75813 


.66523 


.74664! 


.67816 


.73491 


.69088 


.72297 


.70339 


.71080 18 


43 


.65232 


.75794 


.68545 


.74644! 


.67837 


.73472 


.69109 


. 72277 


.70360 


.71059 17 


44 


.65254 


.75775! 


.66566 


.74625 


.67859 


.73452 


.69130 


! 72257 


.70381 


.71039 16 


45 


.65276 


.75756 


.66588 


. 74606 : 


.67880 


.73432 


.69151 


.72236 


.70401 


.71019 


15 


46 


.65298 


.75738! 


.66610 


.74586 


.67901 


.73413 


.69172 


.72216 


.70422 


.70998 


14 


47 


.65320 


.75719 


.66632 


.74567: 


.67923 


.73393 


.69193 


.72196 


.70443 


.70978 


13 


48 


.65342 


.75700 


.66653 


.74548 


.67944 


.73373 


.69214 


.72176 


.70463 


.70957 


12 


49 


.65364 


.75680! 


.66675 


.74528 


.67965 


.73353 


.69235 


.72156 


.70484 


.70937 


11 


50 


.65386 


.75661! 


.66697 


.74509! 


.67987 


.73333 


.69256 


.72136 


.70505 


.70916 


10 


51 


.65408 


.75642 


.66718 


.74489' 


.68008 


.73314 


.69277 


.72116 


.70525 


.70896 


9 


52 


.65430 


.75623' 


.66740 


.74470 


.68029 


.73294 


.69298 


.72095 


.70546 


.70875 


8 


53 


.65452 


.756041 


.66762 


.74451! 


.68051 


.73274 


.69319 


.72075 


.70567 


.70855 


7 


54 


.65474 


.75585 


.66783 


.74431 


.68072 


.732.54 


.69340 


.72055 


.70587 


.70834 


6 


55 


.65496 


. 75566 ! 


.66805 


.74412 


.68093 


.73234 


.69361 


.72035! 


.70608 


.70813 


5 


56 


.65518 


.75547; 


.66827 


.74392 


.68115 


.73215 


.69382 


.72015 


.70628 


.70793 


4 


57 


.65540 


.75528 


.66848 


.74373 


.68136 


.73195 


.69403 


.71995; 


.70649 .70772 


3 


58 


.65562 


.75509; 


.66870 


.74353 


.68157 


.73175 


.69424 


.71974! 


.70670 .70752 


2 


59 


.65584 


.75490; 


.66891 


.74334 


.68179 


.73155 


.69445 


.71954! 


.70690 .70731 


1 


60 


.65606 


.75471 1 
Sine 


.66913 


.74314 


.68200 


.73135 
Sine 


.69466 
Cosin 


.71934 


. 70711 i. 70711 



9 


Cosin 


Cosin i Sine ' 


i Cosin 

1 


Sine 


Cosin | Sine 


49° 


48° 


47° 


46° 


45° 



TABLES. 



6 57 



TABLE VII. 
Natural Tangents and Cotangents. 



"0 


0° 


1° 


2° 


3° 


/ 
00 


Tang 
.00000 


Cotang 


Tang 
.01746 


Cotang 


Tang 
.03492 


Cotang 


Tang 

.05241 


Cotang 
19.0811 


In Unite. 


57.2900 


28.6363 


Jl 


.00029 


3437.75 


.01775 


56.3506 


.03521 


28.3994 


.05270 


IS. 9755 59 


2 


.00058 


1718.87 


.01804 


55.4415 


.03550 


28.1004 


.05299 


18.8711 


58 


3 


.00087 


1145.92 


.01833 


54.5613 


.0:3579 


27.9372 


.05328 


18. 707 8 


57 


4 


.00116 


859.436 


.01862 


53.7086 


.03609 


27.7117 


.05357 


18.0056 


56 


5 


.00145 


687.549 


.01891 


52.8821 


.036^ 


27.4S99 


.05387 


18.5045 


55 


6 


.00175 


572.957 


.01920 


52.0807 


.03667 


27.2715 


.05416 


18.4045 


54 


7 


.00204 


491.106 


.01949 


51.3032 


.03096 


27.0506 


.05445 


18.3655 


53 


8 


.00233 


429.718 


.01978 


50.5485 


.03725 


26.8450 


.05474 


18.2677 


52 


9 


.00262 


381.971 


.02007 


49.8157 


.03754 


26.6367 


.05503 


18.1708 


51 


10 


.00291 


343.774 


.02036 


49.1039 


.03783 


26.4316 


.05533 


18.0750 


50 


11 


.00320 


312.521 


.02066 


48.4121 


.03812 


26.2296 


.05562 


17.9802 


49 


12 


.00349 


286.478 


.02095 


47.7395 


.03842 


26.0307 


.05591 


17.8863 


48 


13 


.00378 


264.441 


.02124 


47.0853 


.03871 


25.8348 


.05620 


17.7934 


47 


14 


.00407 


245.552 


.02153 


46.4489 


.03900 


25.6418 


.05649 


17.7015 


46 


15 


.00436 


229.182 


.02182 


45.8294 


.03929 


25.4517 


.05678 


17.6106 


45 


16 


.00465 


214.858 


.02211 


45.2261 


.03958 


25.2644 


.05708 


17.5205 


44 


17 


.00495 


202.219 


.02240 


44.6388 


.03987 


25.0798 


.05737 


17.4314 


43 


18 


.00524 


190.984 


.02269 


44.0661 


.04016 


24.8978 


.05706 


17.3432 


42 


19 


.00553 


180.932 


.02298 


43.5081 


.04046 


24.7185 


.05795 


17.2558 


41 


20 


.00582 


171.885 


.02328 


42.9641 


.04075 


24.5418 


.05824 


17.1693 


40 


21 


.00611 


163.700 


.02357 


42.4335 


.04104 


24.3675 


.05854 


17.0837 


39 


22 


.OOG40 


156.259 


.02386 


41.9158 


.04133 


24.1957 


.05883 


16.9990 


38 


23 


.00869 


149.465 


.02415 


41.4106 


.04162 


24.0263 


.05912 


16.9150 


37 


24 


.00698 


143.237 


.02444 


40.9174 


.04191 


23.8593 


.05941 


16.8319 


36 


25 


.00727 


137.507 


.02473 


40.4358 


.04220 


23.6945 


.05970 


16.7496 


35 


20 


.00756 


132.219 


.02502 


39.9055 


.04250 


23.5321 


.05999 


16.6681 


34 


27 


.00785 


127.321 


.02531 


39.5059 


.04279 


23.3718 


.00029 


16.5874 


33 


28 


.00815 


122.774 


.02560 


39.8568 


.04308 


23.2137 


.00058 


16.5075 


32 


29 


.00844 


118.540 


.02589 


38.6177 


.04337 


23.0577 


.06087 


16.4283 


31 


30 


.00873 


114.589 


.02619 


38.1885 


.04366 


22.9038 


.06116 


16.3499 


30 


31 


.00902 


110.892 


.02648 


37.7686 


.04395 


22.7519 


.06145 


16.2722 


29 


32 


.00931 


107.426 


.02677 


37.3579 


.04424 


22.6020 


.06175 


16 '. 1952 


23 


33 


.00960 


104.171 


.02706 


36.9560 


.04454 


22.4541 


.06204 


16.1190 


27 


34 


.00989 


101.107 


.02735 


36.5627 


.04483 


22.3081 


.06233 


16.0435 


26 


35 


.01018 


98.2179 


.02704 


36.1776 


.04512 


22.1G40 


.06262 


15.9687 


25 


36 


.01047 


95.4895 


.02793 


35.8006 


.04541 


£2.0217 


.06291 


15.8945 


24 


37 


.01076 


92.9085 


.02822 


35.4313 


.04570 


21.8813 


.06321 


15.8211 


23 


38 


.01105 


90.4633 


.02851 


35.0095 


.04599 


21.7426 


.06350 


15.7483 


22 


39 


.01135 


88.1436 


.02881 


34.7151 


.04628 


21.6056 


.06379 


15.6762 


21 


40 


.01164 


85.9398 


.02910 


34.3078 


.04658 


21.4704 


.06408 


15.6048 


20 


41 


.01193 


83.8435 


.02939 


34.0273 


.04687 


21.3369 


.06437 


15.5340 


19 


42 


.01222 


81.8470 


.029G8 


33.6935 


.04716 


21.2049 


.06467 


15.4638 


18 


43 


.01251 


79.9434 


.02997 


33.3662 


.04745 


21.0747 


.00496 


15.3943 


17 


44 


.01280 


78.1263 


.03026 


33.0452 


.04774 


20.9460 


.06525 


15.3254 


16 


45 


.01309 


76.3900 


.03055 


32.7303 


.04803 


20.8188 


.06554 


15.2571 


15 


46 


.01338 


74.7292 


.03084 


32.4213 


.04S33 


20.6932 


.06584 


15.1893 


14 


47 


.01367 


73.1390 


.03114 


32.1181 


.04862 


20.5691 


.06613 


15.1222 


13 


48 


.01396 


71.6151 


.03143 


31.8205 


.04891 


20.4465 


.06642 


15.0557 


12 


49 


.01425 


70.1533 


.03172 


31.5284 


.04920 


20.3253 


.06671 


14.9898 


11 


50 


.01455 


68.7501 


.03201 


31.2416 


.04949 


20.2056 


.06700 


14.9244 


10 


51 


.01484 


67.4019 


.03230 


30.9599 


.04978 


20.0872 


.06730 


14.8596 


9 


52 


.01513 


66.1055 


.03259 


30.6833 


.05007 


19.9702 


.06759 


14.7954 


8 


53 


.01542 


64.S580 


.03288 


30.4116 1 


.05037 


19.8546 


.06788 


14.7317 


7 


54 


.01571 


63.6567 


.03317 


30.1446 | 


.05066 


19.7403 


.00817 


14.6685 


6 


55 


.01600 


62.4992 


.03346 


29.8823 


.05095 


19.6273 


.06847 


14.6059 


5 


56 


.01629 


61.3829 


.03376 


29.6245 


.05124 


19.5156 


.06876 


14.5438 


4 


57 


.01658 


60.3058 


.0:3405 


29.3711 


.05153 


19.4051 


.06905 


14.4823 


3 


58 


.01637 


59.2659 


.0:3434 


29.1220 J 


.05182 


19.2959 


.01)934 


14.4212 


2 


59 


.01716 


53.2612 


.03463 


28.8771 | 


.05212 


19.1879 


.06963 


14.3607 


1 


60. 


.0174(3 


57.2900 


.03492 


28.6363 j 


.05241 


19.0811 


.00993 14.3007 
Cotang | Tang 



/ 


Cotang 


Tang 


Cotang 


Tang 


Ccl;f»ng 


Tang 


89° 


88° 


87° 


86° 



658 



SUR VE YING. 



TABLE VII.— Continued. 
Natural Tangents and Cotangents. 





4° 


5° 


6° 


7° 




~o 


Tang 
.06993" 


Cotang 


Tang 

.08749 


Cotang J ' 


rang 
10510 


Cotang 


Tang 1 Cotang 


/ 

60 


14.bU07 


11.4301 


9.51436 


.12278 


8.14435 


1 


.07022 


14.2411 


.08778 


11.3919 


10540 


9.48781 


.12308 


8.12481 


59 


2 


.07051 


14.1821 


.08807 


11.3540 


10569 


9.46141 


.12338 


8.10536 


58 


3 


.07080 


14.1235 


.08837 


11.3163 


10599 


9.43515 


.12367 


8.08600 


57 


4 


.07110 


14.0655 


.08866 


11.2789 


10628 


9.40904 


.12397 


8.06674 


56 


5 


.07139 


14.0079 


.08895 


11.2417 . 


10657 


9.38307 


.12426 


8.04756 


55 


6 


.07168 


13.9507 


.08925 


11.2048 . 


10687 


9.35724 


.12456 


8.02848 


54 


7 


.07197 


13.8940 


.08954 


11.1G81 


10716 


9.33155 


.12485 


8.00948 


53 


8 


.07227 


13.8378 


.08983 


11.1316 


10746 


9.30599 


.12515 


7.99058 


52 


9 


.07256 


13.7821 


.09013 


11.0954 . 


10775 


9.28058 


.12544 


7.97176 


51 


10 


.07285 


13.7267 


.09042 


11.0594 . 


10805 


9.25530 


.12574 


7.953C2 


50 


11 


.07314 


13.6719 


.09071 


11.0237 


10834 


9.23016 


.12603 


7.93438 


49 


12 


.07344 


13.6174 


.09101 


10.9882 


10863 


9.20516 


.12633 


7.91582 


48 


13 


.07373 


13.5634 


.09130 


10.9529 


10893 


9.18028 


.12662 


7.89734 


47 


14 


.07402 


13.5098 


.09159 


10.9178 


10922 


9.15554 


.12692 


7.87895 


46 


15 


.07431 


13.4566 


.09189 


10.8829 


10952 


9.13093 


.12722 


7.86064 


45 


16 


.07461 


13.4039 


.09218 


10.8483 


10981 


9.10646 


.12751 


7.84242 


44 


17 


.07490 


13.3515 


.09247 


10.8139 


11011 


9.0S211 


.12781 


7.82428 


43 


18 


.07519 


13.2996 


.09277 


10.7797 


11040 


9.057'89 


.12810 


7.80622 


42 


19 


.07548 


13.2480 


.09306 


10.7457 


11070 


9.03379 


.12840 


7.78825 


41 


20 


.07578 


13.1969 


.09335 


10.7119 


11099 


9.00983 


.12869 


7.77035 


40 


21 


.07607 


13.1461 


.09365 


10.6783 


11128 


8.98598 


.12899 


7.75254 


39 


22 


.07636 


13.0958 


.09394 


10.6450 


11158 


8.9G227 


.12929 


7.73480 


38 


23 


.07665 


13.0458 


.09423 


10.6118 


11187 


8.93867 


.12958 


7.71715 


37 


24 


.07695 


12.9962 


.09453 


10.5789 


11217 


8.91520 


.12988 


7.69957 


36 


25 


.07724 


12.9469 


.09482 


10.5462 


11246 


8.89185 


.13017 


7.68208 


35 


26 


.07753 


12.8981 


.09511 


10.5136 


11276 


8.86862 


.13047 


7.66466 


34 


27 


.07782 


12.8496 


.09541 


10.4813 


.11205 


8.84551 


.13076 


7.64732 


33 


28 


.07812 


12.8014 


.09570 


10.4491 


11335 


8.82252 


.13106 


7.63005 


32 


29 


.07841 


12.7536 


.09600 


10.4172 


11364 


8.79964 


.13136 


7.61287 


31 


30 


.07870 


12.7062 


.09629 


10.3854 


11394 


8.77689 


.13165 


7.59575 


30 


31 


.07899 


12.6591 


.09658 


10.3538 


11423 


8.75425 


.13195 


7.57872 


29 


32 


.07929 


12.6124 


.09688 


10.3224 


11452 


8.73172 


.13224 


7.56176 


28 


33 


.07958 


12.5660 


.09717 


10.2913 


.11482 


8.70931 


.13254 


7.54487 


27 


34 


.07987 


12.5199 


.09746 


10.2602 


11511 


8.68701 


.13284 


7.52806 


26 


35 


.08017 


12.4742 


.09776 


10.2294 


11541 


8.66482 


.13313 


7.51132 


25 


36 


.08046 


12.4288 


.09805 


10.1988 


11570 


8.64275 


.13343 


7.49465 


24 


37 


.08075 


12.3838 


.09834 


10.1683 


11600 


8.62078 


.13372 


7.47806 


23 


38 


.08104 


12.3390 


.09864 


10.1381 


.11629 


8.59893 


.13402 


7.46154 


22 


39 


.08134 


12.2946 


.09893 


10.1080 


11659 


8.57718 


.13432 


7.44509 


21 


40 


.08163 


12.2505 


.09923 


10.0780 


11688 


8.55555 


.13461 


7.42871 


20 


41 


.08192 


12.2067 


.09952 


10.0483 


11738 


i 53402 


.13491 


7.41240 


19 


42 


.08221 


12.1632 


.09981 


10.0187 


.11747 


8.51259 


.13521 


7.39616 


18 


43 


.08251 


12.1201 


.10011 


9.98931 


11777 


8.49128 


.13550 


7.37999 


37 


44 


.08280 


12.0772 


.10040 


9.96007 


11806 


8.47007 


.13580 


7.36389 


36 


45 


.08309 


12.0346 


.10069 


9.93101 


.11836 


8.44896 


.33609 


7.34786 


15 


46 


.08339 


11.9923 


.10099 


9.90211 


11865 


8.42795 


.13639 


7.33190 


14 


47 


.08368 


11.9504 


.10128 


9.87338 


11895 


8.40705 


.13669 


7.31600 


13 


48 


.08397 


11.9087 


.10158 


9.84482 


11924 


8.38625 


.13698 


7.30018 


12 


49 


.08427 


11.8673 


.10187 


9.81641 


11954 


8.36555 


.13728 


7.28442 


11 


50 


.08456 


11.8262 


.10216 


9.78817 


11983 


8.34496 


.13758 


7.26873 


10 


51 


.08485 


11.7853 


.10246 


8.76009 


12013 


8.32446 


.13787 


7.25310 


9 


52 


.08514 


11.7448 


.10275 


9.73217 


12042 


8.30406 


.13817 


7.23754 


8 


53 


.08544 


11.7045 


.10305 


9.70441 


12072 


8.28376 


.13846 


7.22204 


7 


54 


.0S573 


11.6645 


.10334 


9.67680 


12101 


8.26355 


.13876 


7.20661 


6 


55 


.08602 


11.6248 


.10363 


9.64935 


12131 


8.24345 


.13906 


7.19125 


5 


56 


.08632 


11.5853 


.10393 


9.62205 


12160 


8.22344 


.13935 


7.17594 


4 


57 


.08661 


11.5461 


.10422 


9.59490 


12190 


8.20352 


.13965 


7.16071 


3 


58 


.08690 


11.5072 


.10452 


9.56791 


12219 


8.18370 


.13995 


7.14553 


2 


59 


.08720 


11.4685 


.10481 


9.54106 


12249 


8.16398 


.14024 


7.13042 


1 


60 


.08749 


11.4301 


.10510 
Cotang 


9.51436 


12278 


8.14435 


.14054 


7.11537 





Cotang 


Tang 


Tang C 


otang 


Tang 


Cotang 


Tang 


85° 


84° 


83° 


82° 



TABLES. 



659 



TABLE VII.— Continued. 
Natural Tangents and Cotangents. 



"0 




8° 


1 


9° 


Tang 

.14054 


Cotang 


Tang 
.15838 


Cotang 


7.11537 


6.31375 


1 


.14084 


7.10038 


.15868 


6.30189 


2 


.14113 


7.08546 


.15898 


6.29007 


3 


.14143 


7.07059 


.15928 


6.27829 


4 


.14173 


7.05579 


.15958 


6.26655 


5 


.14202 


7.04105 


.15988 


6.25486 


6 


.14232 


7.02637 


.16017 


6.24321 


7 


.14262 


6.91174 


.16047 


6.23160 


8 


.14291 


6.99718 


.16077 


6.22003 


9 


. 14321 


6.98268 


.16107 


6.20851 


10 


.14351 


6.96823 


.16137 


6.19703 


11 


.14381 


6.95385 


.16167 


6.18559 


12 


.14410 


6.93952 


.16196 


6.17419 


13 


.14440 


6.92525 


.16226 


6.16283 1 


14 


.14470 


6.91104 


.16256 


6.15151 


15 


.14499 


6.89688 


.16286 


6.14023 


16 


.14529 


6.88278 


.16316 


6.12899 


17 


.14559 


6.86874 


.16346 


6.11779 1 


18 


.14588 


6.85475 


.16376 


6.10664 


19 


.14618 


6.84082 


.16405 


6.09552 


20 


.14648 


6.82694 


.16435 


6.08444 


21 


.14678 


6.81312 


.16465 


6.07340 


22 


.14707 


6.79936 


.16495 


6.06240 


23 


.14737 


6.78564 


.16525 


6.05143 


24 


.14767 


6.77199 


.16555 


6.04051 


25 


.14796 


6.75838 


.16585 


6.02962 


26 


.14826 


6.74483 


.16615 


6.01878 


27 


.14856 


6.73133 


.16645 


6.00797 


28 


.14886 


6.71789 


.16674 


5.99720 


29 


.14915 


6.70450 


.16704 


5.93046 


30 


.14945 


6.69116 


.16734 


5.97576 


31 


.14975 


6.67787 


.16764 


5.96510 


32 


.15005 


6.66463 


.16794 


5.95448 


33 


.15034 


6.65144 


.16824 


5.94390 


34 


.15064 


6.63831 


.16854 


5.93335 


35 


.15094 


6.62523 


.16884 


5.92283 


36 


.15124 


6.61219 


.16914 


5.91236 


37 


.15153 


6.59921 


.16944 


5.90191 


38 


.15183 


6.58627 


.16974 


5.89151 


39 


.15213 


6.57339 


.17004 


5.88114 


40 


.15243 


6.56055 


.17033 


5.87080 


41 


.15272 


6.54777 


.17063 


5.86051 


42 


.15302 


6.53503 


.17093 


5.85024 


43 


.15332 


6.52234 


.17123 


5.84001 


44 


.15362 


6.50970 


.17153 


5.82982 


45 


.15391 


6.49710 


.17183 


5.81966 


46 


.15421 


6.48456 


.17213 


5.80953 


47 


.15451 


6.47206 


.17243 


5.79944 


48 


.15481 


6.45961 


.17273 


5.78938 


49 


.15511 


6.44720 


.17303 


5.77936 


50 


.15540 


6.43484 


.17333 


5.76937 


51 


.15570 


6.42253 


.17363 


5.75941 


52 


.15600 


6.41026 


.17393 


5.74949 


53 


.15630 


6.39804 


.17423 


5.73960 


54 


.15660 


6.38587 


.17453 


5.72974 


55 


.15689 


6.37374 


.17483 


5.71992 


56 


.15719 


6.36165 


.17513 


5.71013 


57 


.15749 


6.34961 


.17543 


5.70037 


58 


.15779 


6.33761 


.17573 


5.69064 


59 


.15809 


6.32566 


.17603 


5.68094 


GO 


.15838 


6.31375 


.17633 


5.67128 


/ 


Cotang 


Tang 


Cotang 


Tang 




8 


1° 


81 


)° 



10° 



_Tang_ 

.17633 
.17663 
.17693 
.17723 
.17753 
.17783 
.17813 
.17843 
.17873 
.17903 
.17933 

.17963 
.17993 
.18023 
.18053 
.18083 
.18113 
.18143 
.18173 
.18203 
.18233 

.18263 
.18203 
.18323 
.18353 
.18384 
.18414 
.18444 
.18474 
.18504 
.18534 

.18564 

.18594 
.18624 
.18654 
.18684 
.18714 
.18745 
.18775 
.18805 
.18835 

.18865 
.18895 
.18925 
.18955 
.18986 
.19016 
.19046 
.19076 
.19106 
.19136 

.19166 
.19197 
.19227 
.19257 
.19287 
.19317 
.19347 
.19378 
.19408 
.19438 

Cotang 



Cotang 



5.67128 
5.66165 
5.65205 
5.64248 
5.63295 
5.62344 
5.61397 
5.60452 
5.59511 
5.58573 
5.57638 

5.56706 
5.55777 
5.54851 
5.53927 
5.53007 
5.52090 
5.51176 
5.50264 
5.49356 
5.48451 

5.47548 
5.4GG48 
5.45751 
5.44857 
5.43966 
5.43077 
5.42192 
5.41309 
5.40429 
5.39552 

5.38677 
5.37805 
5.36936 
5.36070 
5.35206 
5.34345 
5.33487 
5.32631 
5.31778 
5.30928 

5.30080 
5.29235 
5.28393 
5.27553 
5.26715 
5.25880 
5.25048 
5.24218 
5.2a391 
5.22566 

5.21744 
5.20925 
5.20107 
5.19293 
5.18480 
5.17671 
5.16863 
5.16058 
5.15256 
5.14455 
Tang 



11° 



79 c 



JTang_ 

.19438 
.19468 
.19498 
.19529 
.19559 
.19589 
.19619 
.19649 
.19680 
.19710 
.19740 

.19770 

.19801 
.19831 
.19861 
.19891 
.19921 
.19952 
.19982 
.20012 
.20042 

.20073 
.20103 
.20133 
.20164 
.20194 
.20224 
.20254 
.20285 
.20315 
.20345 

.20376 
.20406 
.20436 
.20466 
.20497 
.20527 
.20557 
.20588 
.20618 
.20648 

.20679 
.26709 
.20739 
.20770 
.20800 
.20830 
.20861 
.20891 
.20921 
.20952 

.20982 
.21013 
.21043 
.21073 
.21104 
.21134 
.21164 
.21195 
.21225 
.21256 



Cotang 



.76595 
.75906 
.75219 
.74534 
.73851 
.73170 
.72490 
.71813 
.71137 
0463 

Cotang Tang 



5.14455 
5.13658 
5.128G2 
5.12069 
5.11279 
5.10490 
5.09704 
5.08921 
5.08139 
5.07360 
5.06584 

5.058?9 
5.0503/ 
5.042,7 
5.03499 
5.02734 
5.01971 
5.01210 
5.00451 
4.99695 
4.98940 

4.98188 
4.97438 
4.96690 
4.95945 
4.95201 
4.94460 
4.93721 
4.92984 
4.92249 
4.91516 

4.90785 
4.90056 
4.89330 
4.88605 
4.878S2 
4.87162 
4.86444 
4.85727 
4.85013 
4.84300 

4.83590 

4.82882 
4.82175 
4.81471 
4.80769 
4.80068 
4.79370 
4.78673 
4.77978 
4.77286 

4. 

4. 
4. 
4 
4. 
4. 
4. 
4. 
4. 
4. 



78° 



60 
59 
58 
57 

5G 
55 
54 
53 

52 
51 
50 

49 
48 
4? 
4G 
45 
4-1 
43 
42 
41 
40 

39 
38 
37 
3G 
35 
34 
33 
32 
31 
30 

29 

28 
27 
2G 
25 
24 
23 
22 
21 
20 

19 

18 
17 
16 
15 
14 
13 
12 
11 
10 

9 

8 
7 
6 
5 
4 
3 
2 
1 




66o 



SURVEYING. 



TABLE VII.— Continued. 
Natural Tangents and Cotangents. 



} 

"6 


12° 


18° 


14 3 


15° 


/ 

60 


Tang 
.21256 


Cotang 


Tang 
.23087 


Cotang 


Tang 

.24933 


Cotang 


Tang 


Cotang 


4.70463 


4.33148 I 


4.01078 


.26795 


3.73205 


1 


.21286 


4.69791 


.23117 


4.32573 


.24964 


4.00582 


.26826 


3.72771 


59 


2 


.21316 


4.69121 


.23148 


4.32001 


.24995 


4.00086 


.26857 


3.72338 


58 


3 


.21347 


4.68452 


.23179 


4.31430 


.25026 


3.99592 


.26888 


3.71907 


57 


4 


.21377 


4.67786 


.23209 


4.30860 


.25056 


3.99099 


.26920 


3.71476 


56 


5 


.21408 


4,67121 


.23240 


4.30291 


.25087 


3.98607 


.26951 


3.71046 


55 


6 


.21438 


4.66458 


.23271 


4.29724 


.25118 


3.98117 


.26982 


3.70616 


54 


7 


.21469 


4.65797 


.23301 


4.29159 


.25149 


3.97627 


.27013 


3.70188 


53 


8 


.21499 


4.65138 


.23332 


4.28595 


.25180 


3.97139 


.270H 


3.69761 


52 


9 


.215^9 


4.64480 


.23363 


4.28032 


.25211 


3.96651 


.27076 


3.69335 


51 


10 


.21560 


4.63825 


.23393 


4.27471 


.25242 


3.96165 


.27107 


3.68909 


50 


11 


.21590 


4.63171 


.23424 


4.26911 


.25273 


3.95680 


.27138 


3.68485 


49 


12 


.21621 


4.62518 


.23455 


4.26352 


.25304 


3.95196 


.27169 


3.6G061 


48 


13 


.21651 


4.61868 


.23485 


4.25795 


.25335 


3.94713 


.27201 


3.676-^8 


47 


14 


.21682 


4.61219 


.23516 


4.25239 


.25366 


3.94232 


.27232 


3.67217 


46 


15 


.21712 


4.60572 


.23547 


4.24685 


.25397 


3.93751 


.27263 


3.G6796 


45 


16 


.21743 


4.59927 


.23578 


4.24132 


.25428 


3.93271 


.27294 


3.66376 


44 


17 


.21773 


4.59283 


.23608 


4.23580 


.25459 


3.92793 


.27326 


3.65C57 


43 


18 


.21804 


4.58641 


.23639 


4.23030 


.25490 


3.92316 


.27357 


3.65538 


42 


19 


.21834 


4.58001 


.23670 


4.22481 


.25521 


3.91839 


.27388 


3.65121 


41 


20 


.21864 


4.57363 


.23700 


4.21933 


.25552 


3.91364 


.27419 


3.64705 


40 


21 


.21895 


4.56726 


.23731 


4.21387 


.25583 


3.90890 


.27451 


3.64289 


39 


22 


.21925 


4.56091 


.23762 


4.20842 


.25614 


3.90417 


.27482 


3.63874 


38 


23 


.21956 


4.55458 


.23793 


4.20298 


.25645 


3.89945 


.27513 


3.63461 


37 


£4 


.21986 


4.54826 


.23823 


4.19756 


.25676 


3.89474 


.27545 


3.63048 


36 


25 


.22017 


4.54196 


.23854 


4.19215 


.25707 


3.89004 


.27576 


3.62636 


35 


26 


.22047 


4.53568 


.23885 


4.18675 


.25738 


3.88536 


.27607 


3.62224 


34 


27 


.22078 


4.52941 


.23916' 


4.18137 


.25769 


3.88068 


.27638 


3.61814 


33 


28 


.22108 


4.52316 


.23946 


4.17600 


.25800 


3.87601 


.27670 


3.61405 


32 


29 


.22139 


4.51693 


.23977 


4.17064 


.25831 


3.87136 


.27701 


3.60996 


31 


30 


.22169 


4.51071 


.24008 


4.16530 


.25862 


3.86671 


.27732 


3 60588 


30 


31 


.22200 


4.50451 


.24039 


4.159*97 


.25893 


3.86208 


.27764 


3.60181 


29 


32 


.22231 


4.49832 


.24069 


4.15465 


.25924 


3.85745 


.27795 


3.59775 


28 


33 


.22261 


4.49215 


.24100 


4.14934 


.25955 


3.85284 


.27826 


3.59370 


27 


34 


.22292 


4.48600 


.24131 


4.14405 


.25986 


3.84824 


.27858 


3.58966 


26 


35 


.22322 


4.47986 


.24162 


4.13877 


.26017 


3.84364 


.27889 


3.58562 


25 


36 


.22353 


4.47374 


.24193 


4.13350 


.26048 


3.83906 


.27921 


3.58160 


24 


37 


.22383 


4.46764 


.24223 


4.12825 


.26079 


3.83449 


.27952 


3.57758 


23 


38 


.22414 


4.46155 


.24254 


4.12301 


.26110 


3.82992 


.27983 


3.57357 


22 


39 


.22444 


4.45548 


.24285 


4.11778 


.26141 


3.82537 


.28015 


3.56957 


21 


40 


.22475 


4.44942 


.24316 


4.11256 


.26172 


3.82083 


.28046 


3.56557 


20 


41 


.22505 


4.44338 


.24347 


4.10736 


-.26203 


3.81630 


.28077 


3.56159 


19 


42 


.22536 


4.43735 


.24377 


4.10216 


.26235 


3.81177 


.28109 


3.55761 


18 


43 


.22567 


4.43134 


.24408 


4.09699 


.26266 


3.80726 


.28140 


3.55364 


17 


44 


.22597 


4.42534 


.24439 


4.09182 


.26297 


3.80276 


.28172 


3.54968 


16 


45 


.22628 


4.41936 


.24470 


4.08666 


.26328 


3.79827 


.28203 


3.54573 


15 


4G 


.22058 


4.41340 


.24501 


4.08152 


.26359 


3.79378 


.28234 


3.54179 


14 


47 


.22689 


4.40745 


.24532 


4.07639 


.28390 


3.78931 


.23206 


3.53785 


13 


48 


.22719 


4.40152 


.24562 


4.07127 


.25421 


3.78485 


.28297 


3.53393 


12 


49 


.22750 


4.39560 


.24593 


4.06616 


.26452 


3.78040 


.28329 


3.53001 


11 


50 


.22781 


4.38969 


.24624 


4.06107 


.26483 


3.77595 


.28360 


3.52609 


10 


51 


.22811 


4.38381 


.24655 


4.05599 


.26515 


3.77152 


.28391 


3.52219 


9 


52 


.22842 


4.37793 


.24686 


4.05092 


.26546 


3.76709 


.28423 


3.51829 


8 


53 


.22372 


4.37207 


.24717 


4.04586 


.26577 


3.76268 


.28454 


3.51441 


7 


54 


.22903 


4.36623 


.24747 


4.04081 


.26608 


3.75828 


.28486 


3.51053 


6 


55 


.22934 


4.36040 


.24778 


4.03578 


.26639 


3.75388 


.28517 


3.50066 


5 


56 


.22964 


4.35459 


.24809 


4.03076 


.26670 


3.74950 


.28549 


3.50279 


4 


57 


.22995 


4.34879 


.24840 


4.02574 


.26701 


3.74512 


.28580 


3.49894 


3 


58 


.23026 


4.34300 


.24871 


4.02074 


.26733 


3.74075 


.28612 


3.49509 


2 


59 


.23056 


4.33723 


.24902 


4.01576 


.26^64 


3.73640 


.28643 


3.49125 


1 


60 

/ 


.23087 


4.33148 


! .24933 


4.01078 


.26795 


3.73205 


.28675 
Cotang 


3.48741 



/ 


Cotang 


Tang 


j Cotang 


Tang 


Cotang 


Tang 


Tang 


77° -4; 


76° 


75° 


74° 



TABLES. 



66 1 



TABLE Mil.— Continued. 
Natural Tangents and Cotangents. 



t 

~0 


16° 


17° 


18° 


19° 


/ 

60 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 
.34433 


Cotang 


.28675 


3.48741 


.30573 


3.27085 


.32492 


3.07768 


2.90421 


1 


.28706 


3.48359 


.30605 


3.26745 


.32524 


3.07464 


.34465 


2.90147 


59 


2 


.28738 


3.47977 


.30637 


3.26406 


.32558 


3.07160 


.34498 


2.89873 


58 


3 


.28769 


3.47596 


.30669 


3.26067 


.32588 


3.06857 


.34530 


2.89600 


57 


4 


.28800 


3.47216 


.30700 


3.25729 


.32621 


3.06554 


.34563 


2.89327 


56 


5 


.28832 


3.46837 


.30732 


3.25392 


.32653 


3.06252 


.34596 


2.89055 


55 


6 


.28864 


3.46458 


.30764 


3.25055 


.32685 


3.05950 


.34628 


2.88783 


54 


7 


.28895 


3.46080 


.30796 


3.24719 


.32717 


3.05649 


.34661 


2.88511 


53 


8 


.28927 


3.45703 


.30828 


3.24383 


.32749 


3.05349 


.34693 


2.88240 


52 


9 


.28958 


3.45327 


.30860 


3.24049 


.32782 


3.05049 


.34726 


2.87970 


51 


10 


.28990 


3.44951 


.30891 


3.23714 


.32814 


3.04749 


.34758 


2 87700 


50 


11 


.29021 


3.44576 


.30923 


3.23381 


.32846 


3.04450 


.34791 


2.87430 


49 


12 


.29053 


3.44202 


.30955 


3.23048 


.32878 


3.04152 


.34824 


2.87161 


48 


13 


.29084 


3.43829 


.30987 


3.22715 


.32911 


3.03854 


.34856 


2.86892 


47 


14 


.29116 


3.43456 


.31019 


3.22384 


.32943 


3.03556 


.34889 


2.86624 


46 


15 


.29147 


3.43084 


.31051 


3.22053 


.32975 


3.03260 


.34922 


2.86356 


45 


16 


.29179 


3.42713 


.31083 


3.21722 


.33007 


3.02963 


.34954 


2.86089 


14 


17 


.29210 


3.42343 


.31115 


3.21392 


.33040 


3.02667 


.34987 


2.85822 


43 


13 


.29242 


3.41973 


.31147 


3.21063 


.33072 


3.02372 


.35020 


2.85555 


42 


19 


.29274 


3.41604 


.31178 


3.20734 


.33104 


3.02077 


.35052 


2.85289 


41 


20 


.29305 


3.41236 


.31210 


3.20406 


.33136 


3.01783 


.35035 


2.85023 


40 


21 


.29337 


3.40869 


.31242 


3.20079 


.33169 


3.01489 


.35113 


2.84758 


39 


22 


.29368 


3.40502 


.31274 


3.19752 


.33201 


3.01196 


.35150 


2.84494 


38 


23 


.29400 


3.40136 


.31306 


3.19426 


.33233 


3.C0903 


.35183 


2.84229 


37 


24 


.29432 


3.39771 


.31338 


3.19100 


.33206 


3.C0611 


.35216 


2.8336:'. 


36 


25 


.29463 


3.39406 


.31370 


3.18775 


.33298 


3.00319 


.35248 


2.83702 


35 


26 


.29495 


3.39042 


.31402 


3.18451 


.33330 


3.00023 


.35281 


2.83439 


34 


27 


.29526 


3.38679 


.31434 


3.18127 


.33363 


2.99738 


.35314 


2.83176 


33 


28 


.29558 


3.38317 


.31466 


3.17804 


.33395 


2.99447 


.35346 


2.82914 


32 


29 


.29590 


3.37955 


.31498 


3.17481 


.83427 


2.99158 


.35379 


2.82653 


31 


30 


.29621 


3.37594 


.31530 


3.17159 


.33460 


2.98868 


.35412 


2.82391 


30 


31 


.29653 


3.37234 


.31562 


3.16838 


.33492 


2.98580 


.35445 


2.82130 


29 


32 


.29685 


3.36875 


.31594 


3.16517 


.33524 


2.98292 


.35477 


2.81870 


28 


33 


.29716 


3.36516 


.31626 


3.16197 


.33557 


2.98004 


.35510 


2.81610 


27 


34 


.29748 


3.36158 


.31658 


3.15877 


.33589 


2.97717 


.35543 


2.81350 


26 


35 


.29780 


3.35800 


.31690 


3.15558 


.33621 


2.97430 


.35576 


2.81091 


25 


36 


.29811 


3.35443 


.31722 


3.15240 


.33654 


2.97144 


.35608 


2.80833 


24 


37 


.29843 


3.3508? 


.31754 


3.14922 


.33686 


2.96858 


.S5641 


2.80574 


23 


38 


.29875 


3.34732 


.31786 


3.14605 


.33718 


2.96573 


.35074 


2.80316 


22 


39 


.29906 


3.34377 


.31818 


3.14288 


.33751 


2.96288 


.35707 


2.80059 


21 


40 


.29938 


3.34023 


.31850 


3.13972 


.33783 


2.96004 


.35740 


2.79802 


20 


41 


.29970 


3.33670 


.31882 


3.13656 


.33816 


2.95721 


.35772 


2.79545 


19 


42 


.30001 


3.33317 


.31914 


3.13341 


.33848 


2.95437 


.35805 


2.79289 


18 


43 


.30033 


3.32965 


.31946 


3.13027 


.33881 


2.95155 


.35838 


2.79033 


17 


44 


.30065 


3.32614 


.31978 


3.12713 


.33913 


2.94872 


.35871 


2.78778 


16 


45 


.30097 


3.32264 


.32010 


3.12400 


.33945 


2.94591 


.35904 


2.78523 


15 


46 


.30123 


3.31914 


.32042 


3.12087 


.33978 


2.94309 


.35937 


2.78269 


14 


47 


.301G0 


3.31565 


.32074 


3.11775 


.34010 


2.94028 


.35909 


2.78014 


13 


40 


.30192 


3.31216 


.32106 


3.11464 


.34043 


2.93748 


.36002 


2.77761 


12 


49 


.30224 


3.30868 


.32139 


3.11153 


.34075 


2.93468 


.36035 


2.77507 


11 


50 


.30255 


3.30521 


.32171 


3.10842 


.34108 


2.93189 


.36068 


2.77254 


10 


51 


.30287 


3.30174 


.32203 


3.10532 


.34140 


2.92910 


.36101 


2.77002 


9 


52 


.30319 


3.29829 


.32235 


3.10223 


.34173 


2.92632 


.36134 


2.76:.-,o 


8 


53 


.30351 


3.29483 


.32267 


3.09914 


.34205 


2.92354 


.36167 


2.76498 


7 


54 


.30382 


3.29139 


.32299 


3.09606 


.34238 


2.92076 


.36199 


2.76247 


6 


55 


.30414 


3.28795 


.32331 


3.09298 


.34270 


2.91799 


.36232 


2. 75996 


5 


56 


.30446 


3.28452 


.32363 


3.08991 


.34303 


2.91523 


.36265 


2.75746 


4 


57 


.30478 


3.28109 


.32396 


3.08685 


.34335 


2.91246 


.36298 


2.75496 


3 


58 


.30509 


3.27767 


.32428 


3.08379 


.34368 


2.90971 


.36331 


2. 152K) 


2 


59 


.30541 


3.27426 


.32460 


3.08073 


.34400 


2.90C96 


.36364 


2.74997 


1 


GO 

/ 


.30573 

Cotang 


3.27085 
Tang 


.32492 


3.07768 


.34433 

Cotang 


2.90421 


.36397 


2.74748 





: Cotang 


Tang 


Tang 


Cotang 


Tang 


r 


7 


3° 


72° 


71° 


1 70° 



662 



SURVEYING. 



TABLE VII.— Continued. 
Natural Tangents and Cotangents. 



~o 


20° 


21° 


22° 


23° 


60 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


.36397 


2.74748 


.38386 


2.60509 


.40403 


2.47509 


.42447 


2.35585 


1 


.36430 


2.74499 


.38420 


2.60283 


.40436 


2.47302 


.42482 


2.35395 


59 


2 


.36463 


2.74251 


.38453 


2.60057 


.40470 


2.47095 


.42516 


2.35205 


58 


3 


.36496 


2.74004 


.38487 


2.59831 


.40504 


2.46888 


.42551 


2.35015 


57 


4 


.36529 


2.73756 


.38520 


2.59606 


.40538 


1 2.46682 


.42585 


2.34825 


56 


5 


.36562 


2.73509 


.38553 


2.59381 


.40572 


2.46476 


.42619 


2.34636 


55 


6 


.36595 


2.73263 


.38587 


2.59156 


.40606 


1 2.46270 


.42654 


2.34447 


54 


7 


.36628 


2.73017 


.38620 


2.58932 


.40640 


2.46065 


.42688 


2.34258 


53 


8 


.36661 


2.72771 


.38654 


2.58708 


.40074 


2.45800 


.42722 


2.34069 


52 


9 


.36694 


2.72526 


.38687 


2.58484 


.40707 


2.45655 


.42757 


2.33881 


51 


10 


.36727 


2.72281 


.38721 


2.58261 


.40741 


2.45451 


.42791 


2.33693 


50 


11 


.36760 


2.72036 


.38754 


2.58038 


.40775 


2.45246 


.42826 


2.33505 


49 


12 


.36793 


2.71792 


.38787 


2.57815 


.40809 


2.45043 


.42860 


2.33317 


48 


13 


.36826 


2.71548 


.33821 


2.57593 


.40843 


! 2.44839 


.42894 


2.33130 


47 


14 


.36859 


2.71305 


.33854 


2.57371 


.4087^ 


2.44636 


.42929 


2.32943 


46 


15 


.36892 


2.71062 


.33888 


2.57150 


.40911 


2.44433 


.42903 


2.32756 


45 


16 


.36925 


2.70819 


.38921 


2.56928 


.40945 


2.44230 


.42998 


2.32570 


44 


17 


.36958 


2.70577 


.33955 


2.56707 


.40979 


2.44027 


.43032 


2.32383 


43 


18 


.36991 


2.70335 


.38988 


2.56487 


.41013 


2.43825 


.43067 


2.32197 


42 


19 


.37024 


2.70094 


.39022 


2.56266 


.41047 


2.43623 


.43101 


2.32012 


41 


20 


.37057 


2.69853 


.39055 


2.56046 


.41081 


2.43422 


.43136 


2.31826 


40 


21 


.37090 


2.69612 


.39089 


2.55827 


.41115 


2.43220 


.43170 


2.31641 


39 


22 


.37123 


2.69371 


.39122 


2.55608 


.41149 


2.43019 


.43205 


2.31456 


38 


23 


.37157 


2.69131 


.39156 


2.55389 


.41183 


2.42819 


.43239 


2.31271 


37 


24 


.37190 


2.68892 


.39190 


2.55170 


.41217 


2.42618 


.43274 


2.31086 


36 


25 


.37223 


2.68653 


.39223 


2.54952 


.41251 


2.42418 


.43308 


2.30902 


35 


26 


.37256 


2.68414 


.39257 


2.54734 


.41285 


2.42218 


.43343 


2.30718 


34 


27 


.37289 


2.68175 


.39290 


2.54516 


.41319 


2.42019 


.43378 


2.30534 


33 


28 


.37322 


2.67937 


.39324 


2.54299 


.41353 


2.41819 


.43412 


2.30351 


32 


29 


.37355 


2.67700 


.39357 


2.54082 


.41387 


2.41620 


.43447 


2.30167 


31 


30 


.37388 


2.67462 


.39391 


2.53865 


.41421 


2.41421 


.43481 


2.29984 


30 


31 


.37422 


2.67225 


.39425 


2.53648 


.41455 


2.41223 


.43516 


2.29801 


29 


32 


.37455 


2.66989 


.39458 


2.53432 


.41490 


2.41025 


.43550 


2.29619 


28 


33 


.37488 


2.66752 


.39492 


2.53217 


.41524 


2.40827 


.43585 


2.29437 


27 


34 


.37521 


2.66516 


.39526 


2.53001 


.41558 


2.40629 


.43620 


2.29254 


26 


35 


.37554 


2.66281 


.39559 


2.52786 


.41592 


2.40432 


.43654 


2.29073 


25 


36 


.37588 


2.66046 


.39593 


2.52571 


.41626 


2.40235 


.43689 


2.28891 


24 


37 


.37621 


2.65811 


.39626 


2.52357 


.41660 


2.40038 


.43724 


2.28710 


23 


38 


.37654 


2.65576 


.39660 


2.52142 


.41694 


2.33841 


.43758 


2.28528 


22 


39 


.37687 


2.65342 


.39694 


2.51929 


.41728 


2.39645 


.43793 


2.28348 


21 


40 


.37720 


2.65109 


.39727 


2.51715 


.41763 


2.39449 


.43828 


2.28167 


20 


41 


.37754 


2.64875 


.39761 


2.51502 


.41797 


2.39253 


.43862 


2.27987 


19 


42 


.37787 


2.64642 


.39795 


2.51289 


.41831 


2.39058 


.43897 


2.27806 


18 


43 


.37820 


2.64410 


.39329 


2.51076 


.41805 


2.38863 


.43932 


2.27626 


17 


44 


.37853 


2.64177 


.39802 


2.50864 


41899 


2.38668 


.43966 


2.27447 


16 


45 


.37887 


2.63945 


.39S96 


•2.50652 


.41933 


2.38473 


.44001 


2.27267 


15 


46 


.37920 


2.63714 


.39930 


2.50440 


.41968 


2.38279 


.44036 


2.27088 


14 


47 


.37953 


2.63483 


.39963 


2.50229 


.42002 


2.38084 


.44071 


2.26909 


13 


48 


.37986 


2.63252 


.39997 


2.50018 


.42036 


2.37891 


.44105 


2. 26730 


12 


49 


.38020 


2.63021 


.40031 


2.49807 


.42070 


2.37697 


.44140 


2.26552 


11 


50 


.38053 


2.62791 


.40065 


2.49597 


.42105 


2.37504 


.44175 


2.26374 


10 


51 


.38086 


2.62561 


.40098 


2.49386 


.42139 


2.37311 


.44210 


2.26196 


9 


52 


.38120 


2.62332 


.40132 


2.49177 


.42173 


2.37118 


.44244 


2.26018 


8 


53 


.38153 


2.62103 


.40166 


2.48967 


.42207 


2.30925 


.44279 


2.25840 


7 


54 


.38186 


2.61874 


.40200 


2.48758 


.42242 


2.36733 


.44314 


2.25663 


6 


55 


.38220 


2.61646 


.40234 


2.48549 


.42276 


2.36541 


.44349 


2.25486 


5 


56 


.38253 


2.61418 


.40267 


2 48340 


.42310 


2.36349 


.44384 


2.25309 


4 


57 


.38286 


2.61190 


.40301 


2.48132 


.42345 


2.36158 


.44418 


2.25132 


3 


58 


.38320 


2.60963 


.40335 


2.47924 


.42379 


2.35967 


.44453 


2.24S56 


2 


59 


.38353 


2.60736 


.40369 


2.47716 


.42413 


2.35776 


.44488 


2 24780 


1 


60 
/ 


.38386 


2.60509 


.40403 
Cotang 


2.47509 


.42447 
Cotang 


2.35585 


.44523 

Cotang 

, 


2.24604 





Cotang 


Tang 


Tang 


Tang 


Tang 


69° 


6 


8° I 


67" 


63° 



TABLES. 



663 



TABLE VII — Continued. 
Natural Tangents and Cotangents. 



/ 
~0 


24° 


I 25° 


26° 


27° 


60 


Tang 
.44523 


Cotang 


1 Tang 
.46631 


Cotang 


1 Tang 
.48773 


Cotang 


Tang 
.50953 


Cotang 


2.24604 


2.14451 


2.05030 


1.96261 


1 


.44558 


2.24428 


.46666 


2.14288 


.48809 


2.04879 


.50989 


1.96120 


59 


2 


.44593 


2.24252 


.46702 


2.14125 


.48845 


2.04728 


.51026 


1.95979 


58 


3 


.44627 


2.24077 


.46737 


2.13963 


.48881 


2.04577 


.51063 


1.95838 


57 


4 


.44662 


2.23902 


.46772 


2.13801 


.48917 


2.04426 ! 


.51099 


1.95698 


56 


5 


.44697 


2.23727 


.46808 


2.13639 


.48953 


2.04276 


.51136 


1.95557 


55 


6 


.44732 


2.23553 


.46843 


, 2.13477 


.48989 


2.04125 | 


.51173 


1.95417 


54 


7 


.44767 


2.23378 


.46S79 


2.13316 


.49026 


2.03975 


.51209 


1.95277 


53 


8 


.44802 


2.23204 


.46914 


2.13154 


.49062 


2.03825 


.51246 


1.95137 


52 


9 


.44837 


2.23030 


.46950 


2.12993 


.49098 


2.03675 


.51283 


1.94997 


51 


10 


.44872 


2.22857 


.46985 


2.12832 


.49134 


2.03526 


.51319 


1.94858 


50 


11 


.44907 


2.22683 


.47021 


2.12671 


.49170 


2.03376 


.51356 


1.94718 


49 


12 


.44942 


2.22510 


.47056 


2.12511 


.49206 


2.03227 


.51393 


1.94579 


48 


13 


.44977 


2.22337 


.47092 


2.12350 


.49242 


2.03078 


.51430 


1.94440 


47 


14 


.45012 


2.22164 


.47123 


2.12190 


.49278 


2.02929 


.51467 


1.94301 


46 


15 


.45047 


2.21992 


.47163 


2.12030 


.49315 


2.02780 


.51503 


1.94162 


45 


16 


.45082 


2.21819 


.47199 


2.11871 


.49351 


2.02631 


.51540 


1.94023 


44 


17 


.45117 


2.21647 


| .47234 


2.11711 


.49387 


2.02483 


.51577 


1.93885 


43 


18 


.45152 


2.21475 


.47270 


2.11552 


.49423 


2.02335 


.51614 


1.93746 


42 


19 


.45187 


2.21304 


.47305 


2.11392 


.49459 


2.02187 


.51051 


1.93608 


41 


20 


.45222 


2.21132 


.47341 


2.11233 


.49495 


2.02039 


.51688 


1.93470 


40 


21 


.45257 


2.20961 


.47377 


2.11075 


.49532 


2.01891 


.51724 


1.93332 


39 


22 


.45292 


2.20790 


.47412 


2.10916 


.495C8 


2.01743 


.51761 


1.93195 


38 


23 


.45327 


2.20619 


.47448 


2.10758 


.49004 


2.01596 


.51798 


1.93057 


37 


24 


.45362 


2.20449 


.47483 


2.10600 


.49640 


2.01449 


.51835 


1.92920 


36 


25 


.45397 


2.20278 


.47519 


2.10442 


.49677 


2.01302 


.51872 


1.92782 


35 


26 


.45432 


2.20108 


.47555 


2.10284 


.49713 


2.01155 


.51909 


1.92645 


34 


27 


.45467 


2.19938 


.47590 


2.10126 


.49749 


2.01008 


.51946 


1.92508 


33 


28 


.45502 


2.19769 


.47626 


2.09969 


.49786 


2.00862 


.51983 


1.92371 


32 


29 


.45538 


2.19599 


.47662 


2.C9811 


.49822 


2.00715 


.52020 


1.92235 


31 


30 


.45573 


2.19430 


.47698 


2.09654 


.49858 


2.00569 


.52057 


1.92098 


30 


31 


.45608 


2.19261 


.47733 


2.09498 


.49894 


2.00423 


.52094 


1.91962 


29 


32 


.45643 


2.19092 


.47769 


2.09341 


.49931 


2.00277 


.52131 


1.91826 


28 


33 


.45678 


2.18923 


.47805 


2.09184 


.49967 


2.00131 


.52168 


1.91690 


27 


34 


.45713 


2.18755 


.47840 


2.09028 


.50004 


1.99986 


.52205 


1.91554 


26 


35 


.45748 


2.18587 


.47876 


2.08872 


.50040 


1.99841 


.52242 


1.91418 


25 


36 


.45784 


2.18419 


.47912 


2.08716 


. 50076 


1.99695 


.52279 


1.91282 


24 


37 


.45819 


2.18251 


.47948 


2.08560 


.50113 


1.99550 


.52316 


1.91147 


23 


38 


.45854 


2.18084 


.47984 


2.08405 


.50149 


1.99406 


.52353 


1.91012 


22 


39 


.45389 


2.17916 


.48019 


2.08250 


.50185 


1.99261 


.52390 


1.90876 


21 


40 


.45924 


2.17749 


.48055 


2.08094 


.50222 


1.99116 


.52427 


1.90741 


20 


41 


.45960 


2.17582 


.48091 


2.07939 


.50258 


1.98972 


.52464 


1.90607 


19 


42 


.45395 


2.17416 


.48127 


2.07785 


.50295 


1.98828 


.52501 


1.90472 


18 


43 


.46030 


2.17249 


.48163 


2.07630 


.50331 


1.98684 


.52538 


1.90337 


17 


44 


.46065 


2.17083 


.48198 


2.07476 


.50368 


1.98540 


.52575 


1.90203 


16 


45 


.46101 


2.10917 


.48234 


2.07321 


.50404 


1.98396 


.52613 


1.90069 


15 


46 


.46136 


2.16751 


.48270 


2.07167 


.50441 


1.98253 


.52650 


1.89935 


14 


47 


.46171 


2.16585 


.48306 


2.07014 


.50477 


1.98110 


.52687 


1.8G801 


13 


48 


.46206 


2.16420 


.48342 


2.C6800 


.50514 


1.97966 


.52724 


1.89667 


12 


49 


.46242 


2.10255 


.48378 


2.06706 


.50550 


1.97823 


.52761 


J. 89533 


11 


50 


.46277 


2.10090 


.48414 


2.06553 


.50587 


1.97681 


.52798 


1.89400 


10 


51 


.46312 


2.15925 1 


.48450 


2.06400 


.50623 


1.97538 


.52836 


1.89266 


9 


52 


.46348 


2.15760 


.48486 


2.06247 


.50660 


1.97395 


.52873 


1.89133 


8 


53 


.46383 


2.15596 


.48521 


2.06094 


.50696 


1.97253 


.52910 


1.89000 


7 


54 


.46418 


2.15432 1 


.48557 


2.05942 


.50733 


1.97111 


.52947 


1.88867 


6 


55 


.46454 


2.15208 


.48593 


2.05790 


.50769 


1.96969 


.52985 


1.88734 


5 


56 


.46489 


2.15104 


.48629 


2.05637 


.50806 


1.96827 


.53022 


1.88602 


4 


57 


.46525 


2.14940 


.48665 


2.05485 


.50843 


1.96685 


.53059 


1.88469 


3 


58 


.46560 


2.14777 


.48701 


2.05333 


.50879 


1.96544 


.53096 


1.88337 


2 


59 


.46595 


2.14614 


.48737 


2.05182 


.50916 


1.96402 


.53134 


1.88205 


1 


60 
/ 


.46631 


2.14451 


.48773 


2.05030 


! .50953 


1.96261 


.53171 
Cotang 


1.88073 



/ 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Tang 


65° 


64° 


63° 


62° 



464 



664 



SUR VE YING. 



TABLE VII.— Continued. 
Natural Tangents and Cotangents. 



t 

~o 


28° 


29° 


30° 


31° 


/ 

60 


Tang 
.53171 


Cotang 


Tang 


Cotang 


Tang 

"^57735" 


Cotang 


Tang 


Cotang 


1.88073 


.55431 


1.80405 


1.73205 ' 


.60086 


1.66428 


1 


.53208 


1.87941 


.55469 


1.80281 


.57774 


1.73089 


.60126 


1.66318 


59 


2 


.53246 


1.87809 


.55507 


1.80158 


.57813 


1.72973 


.60165 


1.66209 


58 


3 


.53283 


1.8T677 


.55545 


1.80034 


.57851 


1.72857 


.60205 


1.66099 


57 


4 


.53320 


1.87546 


.55583 


1.79911 


.57890 


1.72741 


.60245 


1.65990 


56 


5 


.53358 


1.87415 1 


.55621 


1.79788 


1 .57929 


1.72625 


.60284 


1.65881 


55 


6 


.53395 


1.87283 


.55659 


1.79665 


1 .57968 


A. 72509 


.GG'324 


1.65772 


54 


7 


.53432 


1.87152 


.55697 


1.79542 


! .58007 


1.72393 


.60364 


1.65663 


53 


8 


.53470 


1.87021 


.55736 


1.79419 


.58046 


1.72278 


.60403 


1.05554 


52 


9 


.53507 


1.86891 


.55774 


1.79296 


.58085 


1.721G3 


.60443 


1.65445 


51 


10 


.53545 


1.86760 


.55S12 


1.79174 


.58124 


1.72047 


.60483 


1.65337 


50 


11 


.53582 


1.86630 


.55850 


1.79051 


.5S162 


1.71932 


.60522 


1.65228 


49 


12 


.53620 


1X6499 


.55888 


1.78929 


.58201 


1.71817 


.60562 


1.65120 


48 


13 


. 53657 


1.86369 


.55926 


1.78807 


.58240 


1.71702 


.60602 


1.65011 


47 


14 


.53694 


1.86239 


,55964 


1 . 78685 


! .58279 


1.71588 


.60642 


1.64903 


46 


15 


.53732 


1.86109 


.56003 


1.78563 


! .58318 


1.71473 


.60681 


1.64795 


45 


16 


.53769 


1.85979 


.56041 


1.78441 


j .58357 


1.71358 


.60721 


1.64687 


44 


17 


.53807 


1.85850 


.56079 


1.78319 


.58396 


1.712-14 


.60761 


1.64579 


43 


18 


.53844 


1.85720 


.56117 


1.78198 


1 .58435 


1.71129 


.60801 


1.64471 


42 


19 


.53882 


1.C5591 


.56156 


1.78077 


.58474 


1.71015 


.60841 


1.64363 


41 


20 


.53920 


1.85462 


.56194 


1.77955 


.58513 


1.70901 


.60881 


1.64256 


40 


21 


.53957 


1.853&3 


.56232 


1.77834 


.58552 


1.70787 


.60921 


1.64148 


39 


22 


.53995 


1.85204 


.56270 


1.77713 


.58591 


1.70673 


.60960 


1.64041 


38 


23 


.54032 


1.C5075 


.56309 


1.77592 


.58631 


1 .70560 


.61000 


1.63934 


37 


24 


.54070 


1.84946 


.56347 


1.77471 


| .58670 


1.70446 


.61040 


1.63826 


30- 


25 


.54107 


1.84818 


.56385 


1.77351 


.58709 


1.70332 


.61080 


1.63719 


35 


2G 


.54145 


1.84689 


.56424 


1.77230 


.58748 


1.70219 


.61120 


1.63612 


34 


27 


.54183 


1.84561 


.56462 


1.77110 


.58787 


1.70106 


.61160 


1.63505 


33 


28 


.54220 


1.84433 


.56501 


1.76990 


.58826 


1.69992 


.61200 


1.63398 


32 


29 


.5425'3 


1.84305 


.56539 


1.76869 


.58865 


1.C9879 


.61240 


1.63292 


31 


30 


.54286 


1.84177 


.56577 


1.76749 


.58905 


1.69766 


.61280 


1.63185 


30 


31 


.54333 


1.84049 


.56616 


1.76629 


.58944 


1.69653 


.61320 


1.63079 


29 


32 


.54371 


1.83922 


.56654 


1.76510 


.58983 


1.69541 


.61360 


1.62972 


28 


33 


.54409 


1.83794 


.56693 


1.76390 


.59022 


1.69428 


.61400 


1.62866 


27 


34 


.54446 


1.83667 


.56731 


1.76271 


1 .59061 


1.69316 


.61440 


1.62760 


26 


35 


.54484 


1.83540 


.56769 


1.76151 


| .59101 


1.69203 


.61480 


1.62654 


25 


36 


.54522 


1.83413 


.56808 


1.76032 


1 .59140 


1.69091 


.61520 


1.62548 


24 


37 


.54560 


1.83286 


.56846 


1.75913 


1 .59179 


1.68979 


.61561 


1.62442 


23 


38 


.54597 


1.83159 


.56885 


1 75794 


1 .59218 


1.68866 


.61601 


1.62336 


22 


39 


.54635 


1.83033 


.56923 


1.75675 


! .59258 


1.687M 


.61641 


1 . 62230 


21 


40 


.54673 


1.82906 


.56962 


1.75556 


| .59297 


1.68643 


.61681 


1.62125 


20 


41 


54711 


1.82780 


.57000 


1.75437 


.59336 


1.68531 


.61721 


1.62019 


19 


42 


.54748 


1.82654 


.57039 


1.75319 


.59376 


1.68419 


.61761 


1.61914 


18 


43 


.54786 


1.82528 


.57078 


1.75200 


.59415 


1.68308 


.61801 


1.61808 


17 


44 


.54824 


1.82402 


.57116 


1.75082 


.59454 


1.68196 


.61842 


1.61703 


16 


45 


.54862 


1.82276 


.57155 


1.74964 


.59494 


1.68085 


.61882 


1.61598 


15 


46 


.54900 


1.82150 


.57193 


1.74846 


| .59533 


1.67974 


.61922 


1.61493 


14 


47 


.54938 


1.82025 


.57232 


1.74728 


.59573 


1.67863 


.61902 


1.61388 


13 


48 


.54975 


1.81899 


.57271 


1.74610 


.59612 


1 . 67752 


.62003 


1.61283 


12 


49 


.55013 


1.81774 


.57309 


1.74492 


; .59651 


1.67641 


.62043 


1.61179 


11 


50 


.55051 


1.81649 


.57348 


1 . 74375 


I .59691 


1.67530 


.62083 


1.61074 


10 


51 


.55089 


1.81524 


.57386 


1.74257 


i .59730 


1.67419 


.62124 


1.60970 


9 


52 


.55127 


1.81399 


.57425 


1.74140 


! .59770 


1.67309 


.62164 


1.60865 


8 


53 


.55165 


1.81274 


.57464 


1.74022 


.59809 


1.67198 


.62204 


1.60761 


7 


54 


.55203 


1.81150 


.57503 


1.73905 


I .59849 


1.67088 


.62245 


1.60657 


6 


55 


.55241' 


1.81025 


.57541 


1.73788 


.59888 


1.66978 


.62285 


1.60553 


5 


56 


.55279 


1.80901 


.57580 


1.73671 


.59928 


1.66867 


.62325 


1 . 60449 


4 


57 


.55317 


1.80777 


.57619 


1.73555 


.59967 


1.66757 


.62366 


1.60345 


3 


58 


.55355 


1.80653 


.57657 


1.73438 


.60007 


1.66647 


.62406 


1.60241 


2 


59 


.55393 


1.80529 


.57696 


1.73321 


.60046 


1.66538 


.62446 


1.60137 


1 


60 


.55431 
Cotang 


1 . N0405 


.57735 
Cotang 


1.73205 


.60086 
Cotang 


1.66428 


.62487 
Cotang 


1.60033 
Tang 



/ 

— ■ 


Tang 


Tang 


Tang 


6 


1° 


6 


0° 


59° 


58° 



TABLES. 



665 



TABLE VII. — Continued. 
Natural Tangents and Cotangents. 



9 
~0 


32° 


33° 


34° 


35° 


00 


Tang 
.62487 


Cotang 
1.60033 


Tang 
.64941 


Cotang 


Tang | 
.07451 


Cotang 


Tang 
.70021 


Cotang 


1.53986 


1.48256 


1.42815 


1 


.62527 


1.59930 


.64982 


1.53888 


.67493 


1.48163 


.70064 


1.42720 


59 



lb 


.62508 


1.59826 


.65034 


1.53791 


.67536 


1 48070 


.70107 


1.42638 


58 


3 


.62608 


1.59723 


.65005 


1.53693 


.67578 


1.47977 


.70151 


1.42550 


57 


4 


.62649 


1.59620 


.65100 


1 . 53595 


.670^0 


1.47885 


.70194 


1.42462 


56 


5 


.62689 


1.59517 


.65148 


1.53497 


.67003 


1.47792 


.70238 


1.42374 


55 


6 


.62730 


1.59414 


.65189 


1.53400 


.67705 


1.47099 


.70281 


1.42286 


54 


7 


.62770 


1.59311 


.65231 


1.53302 


.67748 


1.47607 


.70025 


1.42198 


53 


8 


.62811 


1.59208 


.65272 


1.53205 


.67790 


1.47514 


.70368 


1.42110 


52 


9 


.62852 


1.59105 


.65314 


1.53107 


.67832 


1.47422 


.70412 


1.42C22 


51 


10 


.C2892 


1.59002 


.65355 


1.53010 


.07875 


1.47330 


. 70455 


1.41934 


50 


11 


.62933 


1.53900 


.65397 


1.52913 


.67917 


1.47238 


.70499 


1.41847 


49 


12 


.629:3 


1.58797 


.65438 


1.52816 


.67900 


1.47146 


.70542 


1.41759 ! 


^3 


13 


.63014 


1.58095 


.65480 


1.52719 


.68002 


1.47053 


.70586 


1.41672 


£-7 


14 


.63055 


1.58593 


.65521 


1.52022 


.63045 


1.4G9G2 


.70G23 


1.41534 


40 


15 


.63035 


1.58490 


. 65563 


1.52525 


.08088 


1.46870 


.70673 


1.41497 


•iO 


16 


.63136 


1.58388 


.65004 


1.52429 


.68130 


1.4G778 


.70717 


1.41409 


41' 


17 


.63177 


1.58280 


.65646 


1.52332 


.63173 


1.46686 


.70700 


1.41322 


4g 


18 


.63217 


1.58184 


.65688 


1.52235 


.00015 


1.4G595 


.70304 


1.41235 


42 


19 


.63253 




.05729 


1.52139 


.03258 


1.4C5C3 


.10043 


1.41148 


41 


20 


.632'J9 


1.57981 


. C5771 


1.52043 


.C8301 


1.40411 


.70891 


1.41061 


40 


21 


.63340 


1.57879 


.05S13 


1.51946 


.GS343 


1.4G320 


.70005 


1.40971 


S9 


22 


.633S0 


1'. 57773 


.05354 


1.51850 


.68386 


1.4G229 


.70379 


1.40837 


33 


23 


.63421 


1.57670 


.05390 


1.51754 


. 68429 


1.40137 


.71023 


1.40800 


37 


24 


.63403 


1 . 57575 


.05X3 


1.51C53 


.03471 


1.40046 


.71000 


1.40714 


36 


25 


.635C3 


1.57474 


.659S0 


1.515C2 


J00514 


1.45955 


.71110 


1.40627 


35 


26 


.C3544 


1.57372 


.60021 


1.514C6 


.08557 


1.45S84 


.71154 


1.40540 


04 


27 


.63584 


1.57271 


.6G0G3 


1.51370 


.C8C0O 


1.45773 


.71193 


1.40454 


33 


28 


.63625 


1.57170 


.GG105 


1.51215 


.G3G42 


1.450:2 


.71242 


1.40307 


03 


29 


.03606 


1.570G9 


.00147 


1.51119 


.030:5 


1.45592 


.71285 


1.40281 


31 


30 


.63707 


1.5G9G9 


.CG1C3 


1.510S4 


.03723 


1.45501 


.71329 


1.40195 


20 


31 


.C3743 


1.56868 


.60239 


1.50988 1 


.03771 


1.45410 


.71373 


1.40109 


29 


32 


.C3789 


1.50707 


.66212 


1.50393 


.C8314 


1.45320 


.71417 


1.4O022 


23 


33 


.03830 


1.50067 


.66314 


1.50797 i 


.03157 


1.45229 


.71401 


1.39938 


27 


34 


.63871 


1.56566 


.00X0 


1.50702 1 


.C3000 


1.45139 


.71505 


1.39850 


28 


35 


.63912 


1.56406 


.60393 


1.50607 


r 20 ' 


1.45049 


.71549 


1.39764 


25 


36 


.63953 


1.56300 


! .00440 


1.50512 1 


.68985 


1.44958 


.71593 


1.39079 


24 


37 


.63994 


1.56265 


.00482 


1.50417 


.G9023 


1.44808 


.71637 


1.39593 


23 


38 


.C4035 


1.56165 


.66524 


1.50322 1 


.C3071 


1.44778 


.71081 


1.39507 122 


39 


.64076 


1.560C5 


.06500 


1.50228 


.C9114 


1.44GS8 


.71125 


1.33421 ;21 


40 


.64117 


1.55906 


.00008 


1.50133 


.09157 


1.44598 


.71709 


1.3930G 


CO 


41 


.64158 


1.55866 


.GGG50 


1.50038 


.69200 


1.44508 


.71313 


1.09250 


is 


42 


.64199 


1.55700 


.00092 


1.49314 


.00313 


1.44418 


.71857 


1.33165 


13 


43 


.64240 


1.55006 


.06734 


1.49849 


.69336 


1.44039 


.71901 


1.33079 


17 


44 


.04261 


1.55507 


.00716 


1.49755 


.03039 


1.44239 


.71946 


1.00094 


16 


45 


.64323 


1.55467 


.00818 


1.49001 


.69372 


1.44149 


.71990 


1.38909 


15 


46 


.64303 


1.5536S 


: .60800 


1.49503 


.69410 


1.44060 


.73004 




14 


47 


.64404 


1.55269 


! .00902 


1.49472 


.09459 


1.43970 


.7207S 


1.38733 


13 


48 


.644 56 


1.55170 


.60944 


1.49318 


.09502 


1.43881 


.72122 


1.38653 


12 


49 


.04^37 


1.55071 


.60986 


1.49284 


.69545 


1.43792 


.72107 


1.38568 


H 


50 


.64528 


1.54972 


.67028 


1.49190 


.09588 


1.43703 


.72211 


1.38484 


|10 


51 


.64569 


1.54873 


.67071 


1.49097 


.69631 


1.43614 


.73255 


1.38399 


9 


52 


.64010 


1.54774 


.67113 


1.49003 


.09075 


1.43525 


. 12399 


1.38314 


8 


53 


.64052 


1.54075 


.67155 


1.48909 


.69713 


1.43436 


.72344 


1.38229 


7 


54 


.64093 


1.54576 


.07197 


1.48310 


.69701 


1.43347 


.72388 


1.38145 


6 


55 


.64734 


1.54418 


.67239 


1.48722 


.69804 


1.43258 


.72432 


1.38060 


5 


5e 


. 6471 5 


1.54379 


.67282 


1.48629 


.09847 


1.43109 


.72177 


1.37976 


4 


5' 


.64817 


1.54281 


i .67324 


1.48530 


.69891 


1.43080 


.72521 


1.37891 


3 


58 


.64358 


1.54183 


.67306 


1.48442 


.69934 


1.42992 


.72505 


1.37807 


2 


5J 


1 .64 S99 


1.540S5 


.67409 


1.48349 


.09977 


1.42903 


i .72010 


1.37722 


1 


6( 
/ 


) .64941 1.53986 


.67451 


1.4835(5 


.70021 


1.42815 


' .72654 
Cotang 


1.37638 





Cotang | Tang 


Cotang 


| Tang 


Cotang 


1 Tang 


Tang 


57° 




56° 


55° 


54° 



666 



SUR VE YING. 



TABLE VII. — Continued. 
Natural Tangents and Cotangents. 



— 

~0 


36° 


37° 


38° 


39° 


60 


Tang 1 Cotang 


Tang 
. 75355 


Cotang 


Tang 


Cotang 


Tang | 
.80978 


Cotang 
1.23490 


.72654 


1.37638 


1.32704 


.78129 


1.27994 


I 


.72699 


1.37554 


.75401 


1.32624 


.78175 


1.27917 


.81027 


1.23416 


59 


2 


.72743 


1.37470 


.75447 


1.32514 


.78222 


1.27841 


.81075 


1.23343 


58 


3 


.72788 


1.37386 


.75402 


1.32464 


.78269 


1.27764 


.81123 


1.23270 


57 


4 


.72832 


1.37302 


.75533 


1.32384 


.78316 


1.27688 


.81171 


1.23196 


56 


5 


.72877 


1.37218 


.75584 


1.32304 


.78363 


1.27611 


.81220 


1.23123 


55 


6 


.72921 


1.37134 


.75629 


1.32224 


.78410 


1.27535 


.81268 


1.23050 


54 


7 


.72966 


1.37050 


.75675 


1.32144 


.78457 


1.27458 


.81316 


1.22977 


53 


8 


.73010 


1.36967 


.75721 


1.32064 


.78504 


1.27382 


.81364 


1.22904 


52 


9 


.73055 


1.36883 


.75767 


1.31 984 


.78551 


1.27306 


.81413 


1.22831 


51 


10 


.73100 


1.36800 


.75812 


1.31904 


.78598 


1.27230 


.81461 


1.22758 


50 


11 


.73144 


1.36716 


.75858 


1.31825 


.78645 


1.27153 


.81510 


1.22685 


49 


12 


.73189 


1.36633 


.75904 


1.31745 


.78692 


1.27077 


.81558 


1.22612 


48 


13 


.73234 


1.36549 


.75950 


1.31666 


.73739 


1.27001 


.81000 


1.22539 


47 


14 


.73278 


1.36466 


.75996 


1.31586 


.73786 


1.26925 


.81655 


1.22467 


46 


15 


.13323 


1.36383 


.76042 


1.31507 


.78834 


1.26849 


.81703 


1.22394 


45 


16 


.73368 


1.36300 


.76088 


1.31427 


.78881 


1.26774 


.81752 


1.22321 


44 


17 


.73413 


1.33217 


.76134 


1.31348 


.73923 


1.2GG98 


.81C00 


1.22249 


43 


18 


.73457 


1.36134 


.701C0 


1.31239 


.73975 


1.23622 


.81849 


1.22176 


42 


19 


.73502 


1.33051 


.76226 


1.31130 


.79022 


1.26546 


.81898 


1.22104 


41 


20 


.73547 


1.35963 


.76272 


1.31110 


.79070 


1.2G471 


81946 


1.22031 


40 


21 


.73592 


1.35835 




1.31031 


.79117 


1.26395 


.81005 


1.21959 


39 


22 


. 73637 


1.35002 


.73304 


1.30052 


.70104 


1.26319 


.82344 


1.21886 


33 


23 


.73681 


1.35719 


.73410 


1.30373 


.79212 


1.26244 


.82092 


1.21814 


37 


24 


.73726 


1.35037 


.73456 


1.30795 


.70259 


1.26169 


.82141 


1.21742 


30 


25 


.73771 


1.35554 


.76502 


1.307f6 


.70306 


1.26093 


.82100 


1.21670 


35 


26 


.738^6 


1.35472 


.73518 


1.30037 


.79354 


1.20018 




1.21598 


84 


27 


.73861 


1.353G9 


.76534 


1 . 30558 


.79401 


1.25943 


.62287 


1.21526 


33 


28 


.73006 


1.35307 


.73340 


1.30430 


.79449 


1.25C67 


.82336 


1.21454 


32 


29 


.73951 


1.35224 


.73386 


1.30401 


.70496 


1.25702 


823C5 


1.21382 


31 


30 


.7399 J 


1.35142 


.76733 


1.30323 


.79544 


1.25717 


! 62434 


1.21310 


30 


31 


.74041 


1.S5060 


.76779 


1.20244 


.79591 


1.25642 


.62483 


1.21238 


29 


32 


.74033 


1.34078 


.7'0C~5 


1.30106 


.70639 


1.25567 


.82531 


1.21106 


23 


33 


.74131 


1.34806 


.76871 


1. 30037 


.70336 


1.25492 


.82560 


1.21094 


27 


34 


.74176 


1.34814 


.76913 


1.30009 


.70734 


1.25417 


.82029 


1.21023 


26 


35 


.74221 


1.34732 


.769C4 


1.23931 


.70; 31 


1.25343 


.£2078 


1.20951 


25 


36 


.74267 


1.34650 


.77010 


1.29G53 


.79829 


1.25268 


.62727 


1.20879 


24 


37 


.74312 


1.34568 


.77057 


1.23775 


.79877 


1.25193 


.82776 


1.20808 


23 


38 


.74357 


1.34487 


.771C3 


1.20C06 


.79924 


1.25118 


.82825 


1.20736 


22 


39 


.74402 


1.34405 


.77149 


1.20018 


.79972 


1.25044 


.82874 


1.20GC5 


21 


40 


.74447 


1.34323 


.77193 


1.29541 


.80020 


1.24969 


.82923 


1.20533 


20 


41 


.74492 


1.34242 


.77242 


1.29463 


.80067 


1.24895 


.82972 


1.20522 


19 


42 


.74533 


1.341G0 


! 77283 


1.20385 


.80115 


1.24820 


. 83022 


1.20451 


18 


43 


.74583 


1.34079 


.77335 


1.20307 


.801C3 


1.24746 


! 63071 


1.20379 


17 


44 


.74028 


1.33998 


.77332 


1.29229 


.80211 


1.24072 


.83120 


1.20308 


16 


45 


.74674 


1.33916 


.77423 


1.23152 


.80258 


1.24597 


.881C9 


1.20237 


15 


46 


.74719 


1.33835 


. 77475 


1.29074 


.80306 


1.24523 


.63218 


1.20166 


14 


47 


.74764 


1.33754 


.77521 


1.23997 


.80354 


1.24449 


. Lo/vCy 


1.20095 


13 


48 


.74810 


1.33673 


.77568 


1.23919 


.80402 


1.24375 


.63317 


1.20024 


12 


49 


.74855 


1.33592 


.77C15 


1.28842 


.80450 


1.24C01 


.833C6 


1.19953 


11 


50 


.74900 


1.33511 


.77661 


1.28764 


.80498 


1.24227 


.83415 


1.19882 


10 


51 


.74946 


1.33430 


.77708 


1.28687 


.80546 


1.24153 


.83465 


1.19811 


9 


52 


.74991 


1.33349 


.77754 


1.28610 


.80594 


1.24079 


.83514 


1.19740 


8 


53 


.75037 


1.33268 


.77801 


1.28533 


.80642 


1.24005 


.83504 


1.19CG9 


7 


54 


.75082 


1.33187 


.77848 


1.28456 


.80690 


1.23931 


.83613 


1.19599 


6 


55 


.75128 


1.33107 


.77895 


1.28379 


.80738 


1.23858 


.83062 


1.19528 


5 


56 


.75173 


1.33026 


.77941 


1.28302 


.80786 


1.23784 


.83712 


1.19457 


4 


5? 


.75219 


1.32946 


.77988 


1.28225 


.80834 


1.23710 


.83701 


1.19387 


3 


58 


.75264 


1.32865 


.78035 


1.28148 


.80882 


1.23637 


.83811 


1.19316 


2 


59 


.75310 


1.32785 


.78082 


1.28071 


.80930 


1.23563 


.83860 


1.19246 


1 


60 
f 


.75355 
Cotang 


1.32704 


.78129 


1.27994 


.80978 


1.23490 


.83910 


1.19175 



/ 


Tang 


Cotang 


Tang 


i Cotang 


Tang 


Cotang 


Tang 


53° 


1 52° 


51° 


1 t 


J 



TABLES. 



667 



TABLE VII.— Continued. 
Natural Tangents and Cotangents. 



"0 


40° 


41° 


42° 


43° 


/ 

60 


Tang | 


Cotang 


Tang 
.86929 


Cotang 


Tang | 
.90040 


Cotang 


Tang 

.93252 


Cotang 


.83910 


1.19175 


1.15037 


1.11061 


1.07237 


1 


.83960 


1.19105 


.86980 


1.14969 


.90093 


1.10996 


.93306 


1.07174 


59 


2 


.84009 


1.19035 


.87031 


1.14902 


.90146 


1.10931 


.93S60 


1.07112 


58 


3 


.84059 


1.18964 


.87082 


1.14834 


.90199 


1.10867 


.93415 


1.07049 


57 


4 


.84108 


1.18894 


.87133 


1.14767 


.90251 


1.10802 


.93469 


1.06987 


56 


5 


.84158 


1.18824 


.87184 


1.14699 


.90304 


1.10737 


.93524 


1.06925 


55 


6 


.84208 


1.18754 


.87236 


1.14632 


.90357 


1.10672 


.93578 


1.06862 


54 


7 


.84258 


1.18684 


.87287 


1.14565 


.90410 


1.10607 


.93633 


1.06800 


53 


8 


.81307 


1.18614 


.87338 


1.14498 


.90463 


1.10543 


.93688 


1.06738 


52 


9 


.84357 


1.18544 


.87389 


1.14430 


.90516 


1.10478 


.93742 


1.06676 


51 


10 


.84407 


1.18474 


.87441 


1.14363 


.90569 


1.10414 


.93797 


1.06613 


50 


11 


.84457 


1.18404 


.87492 


1.14296 


.90621 


1.10349 


.93852 


1.06551 


49 


12 


.84507 


1.18334 


.87543 


1.14229 


.90674 


1.10285 


: .93906 


1.06489 


43 


13 


.84556 


1.18264 


.87595 


1.14162 


.90727 


1.10220 


I .93961 


1.06427 


47 


14 


.84606 


1.18194 


.87646 


1.14095 


.90781 


1.10156 


1 .94016 


1.06365 


46 


15 


.84658 


1.18125 


.87698 


1.14028 


.90834 


1.10091 


; .940;i 


1.06303 


45 


16 


.84706 


1.18055 


.87749 


1.13961 


.90887 


1.10027 


1 .94125 


1.06241 


44 


17 


.84756 


1.17986 


.87801 


1.13894 


.90940 


1.009C3 


1 .94180 


1.C6179 


43 


18 


.84806 


1.17916 


.87852 


1.13828 


.90993 


1.09809 


\ .94235 


1.C6117 


42 


19 


.84856 


1.17846 


.87904 


1.13761 


.91046 


1.09834 


1 .942C0 


1.C6056 


41 


20 


.84906 


1.17777 


.87955 


1.13694 


.91099 


1.09770 


.04345 


1.05994 


40 


21 


.84956 


1.17708 


.88007 


1.13627 


.91153 


1.09706 


.94400 


1.05932 


39 


22 


.85006 


1.17638 


.88059 


1.13561 


.91206 


1.09642 


.94455 


1.05870 


33 


23 


.85057 


1.17569 


.88110 


1.1:3494 


.91259 


1.09578 


.94510 


1.05809 


37 


24 


.85107 


1 . 17500 


.88102 


1.13428 


.91313 


1.09514 


.94565 


1.05747 


36 


25 


.85157 


1.17430 


.88214 


1.13361 


.91-366 


1.09450 


.94620 


1.05685 


35 


2G 


.85207 


1.17361 


.88205 


1.13295 


.91419 


1.09386 


.94676 


1.05624 


34 


27 


.85257 


1.17292 


.88317 


1.13223 


.91473 


1.09322 


.94731 


1.05562 


33 


28 


.85308 


1.17223 


.88369 


1.13162 


.91526 


1.09258 


.94786 


1.05501 


32 


29 


.85358 


1.17154 


.88421 


1.13096 


.91580 


1.09195 


.94841 


1.05439 


31 


30 


.85408 


1.170S5 


.88473 


1.13029 


.91633 


1.09131 


.94896 


1.05378 


30 


31 


.85458 


1.17016 


.88524 


1.12963 


.91087 


1.09067 


.949.-2 


1.05317 


29 


32 


.85509 


1.16947 


.88576 


1.12897 


.01740 


1.09003 


.05007 


1.05255 


28 


33 


.855:,9 


1.16878 


.88628 


1.12831 


.91794 


1.0S940 


.95062 


1.05194 


27 


34 


.85609 


1.16S03 


.88680 


1.12765 


.91847 


1.08876 


.95118 


1.05133 


26 


35 


.85060 


1.10741 


.88732 


1.12699 


.91901 


1.08813 


.95173 


1.05072 


25 


3G 


.85710 


1.16672 


.88784 


1.12633 


.91955 


1.08749 


.95229 


1.05010 


24 


37 


.85761 


1.16603 


.88836 


1.12567 


.92008 


1.08686 


.95284 


1.04949 


23 


38 


.85811 


1.10535 


.88888 


1.12501 


.92062 


1.08622 


.95340 


1.04888 


22 


39 


.85862 


1.16466 


.88940 


1.12435 


.92116 


1.08559 


.95395 


1.04827 


21 


40 


.85912 


1.16398 


.88992 


1.12369 


.92170 


1.08496 


.95451 


1.04766 


20 


41 


.859C3 


1.16329 


.89045 


1.12303 


.92224 


1.08432 


.95506 


1.04705 


19 


42 


.86014 


1.16261 


.89097 


1.12238 


.92277 


1.08369 


.95562 


1.04644 


18 


43 


.86004 


1.16192 


.89149 


1.12172 


.92331 


1.08306 


.95618 


1.04583 


17 


44 


.86115 


1.16124 


.89201 


1.12106 


.92385 


1.08243 


.95673 


1.04522 


16 


45 


.86166 


1.16056 


.89253 


1.12041 


.92439 


1.08179 


.95729 


1.04461 


15 


46 


.86216 


1.15987 


.89306 


1.11975 


.92493 


1.08116 


.95785 


1.04401 


14 


47 


.86267 


1.15919 


.89358 


1.11909 


.92547 


1.08053 


.95841 


1.04340 


13 


48 


.86318 


1.15851 


.89410 


1.11844 


.92601 


1 .07990 


.95897 


1.04279 


12 


49 


.86368 


1 . 15783 


.89463 


1 11778 


.92655 


1.07927 


.95952 


1.04218 


11 


50 


.86419 


1.15715 


.89515 


1.11713 


.92709 


1.07864 


.96008 


1.G4158 


10 


51 


.86470 


1.15647 


.89567 


1.11648 


.92763 


1.07801 


.96064 


1.04097 


9 


52 


.86521 


1.15579 


.89620 


1.11582 


.92817 


1.07738 


.96120 


1.04036 


8 


53 


.86572 


1.15511 


.89672 


1.11517 


.92872 


1.07676 


.96176 


1.03976 


7 


54 


.86623 


1.15443 


.89725 


1.11452 


.92926 


1.07613 


.96232 


1.03915 


6 


55 


.86674 


1.15375 


.89777 


1.11387 


.92980 


1.07550 


.96288 


1.03855 


5 


56 


.86725 


1.15308 


.89830 


1.11321 


.93034 


1.07487 


.96344 


1.03794 


4 


57 


.86776 


1.15240 


.89883 


1.11256 


.93088 


1.07425 


.96400 


1.03734 


3 


58 


.86827 


1.15172 


.899% 


1.11191 


.93143 


1.07362 


.96457 


1.03674 


2 


59 


.86878 


1.15104 


.89988 


1.11126 


.93197 


1.07299 


.96513 


1.03613 


1 


60 

/ 


.86929 
Cotang 


1.15037 


.90040 


1.11061 


.93252 


1.07237 


.96569 
Cotang 


1.03553 
Tang 



1 


Tang 


Cotang 


Tang 


Cotang 


Tang 


49° 


48° 


1 47° 


! 46° 



66S 



SUR VE YING. 



TABLE Nil.— Continued. 
•Natural Tangents and Cotangents. 





44° 


, 




44° 




1 ( 


44° 






Tang 


Cotang 




20 


Tang 
.97700 


Cotang 




40 


Tang 

.98843 


Cotang 







.96569 


1.03553 


60 


1.02355 


40 


1.01170 


20 


1 


.96625 


1.03493 


59 


21 


.97756 


1.02295 


39 


41 


.98901 


1.01112 


19 


2 


.96681 


1.03433 


58 


22 


.97813 


1.02236 


38 


42 


.98958 


1 01053 


18 


3 


.96738 


1.03372 


57 


23 


.97870 


1.02176 


37 


43 


.99016 


1.00994 


17 


4 


.93791 


1.03312 


56 


24 


.97927 


1.02117 


36 


44 


. 99073 


1.00935 


16 


5 


.90850 


1.G3252 


55 


25 


.97984 


1.02057 


35 


45 


.99131 


1.00876 


15 


6 


.96907 


1.03192 


54 


26 


.98041 


1.01998 


34 


46 


,99189 


1.U0818 


14 


7 


.96963 


1.03132 


53 


27 


.98098 


1.01939 


33 


47 


.99247 


1.00759 


13 


8 


.97020 


1 .03072 


52 


28 


.98155 


1.01879 


32 


48 


.99304 


1.00701 


12 


9 


.97076 


1.03012 


51 


29 


.98213 


1.01820 


31 


49 


.99362 


1.00642 


11 


10 


.97133 


1.02952 


50 


30 


.98270 


1.01761 


30 


50 


.99420 


1.00583 


10 


11 


.97189 


1.02892 


49 


31 


.98327 


1.01702 


29 


51 


.99478 


1.00525 


9 


12 


.97246 


1.02832 


48 


32 


.98384 


1.01642 


28 


52 


.99536 


1.00467 


8 


13 


.97302 


1 .02772 


47 


33 


.98441 


1.01583 


27 


53 


.99594 


1.00408 


7 


14 


.97359 


1.02713 


46 


34 


.98499 


1.01524 


26 


54 


.99652 


1.00350 


6 


15 


.97416 


1.02653 


45 


35 


.98556 


1.01465 


25 


55 


.99710 


1.00291 


5 


16 


.97472 


1.02593 


44 


36 


' .98613 


1.01406 


24 


56 


.99768 


1.00233 


4 


1? 


.97529 


1.02533 


43 


37 


.98671 


1.01347 


2i 


57 


.99826 


1.00175 


3 


18 


.97586 


1.02474 


42 


38 


.98728 


1.01288 


22 


58 


.99884 


1.00116 


2 


19 


.97643 1.02414 


41 


39 


. 98786 


1.01229 


21 


59 


.99942 


1.00058 


1 


20 


.97700 1.02355 


40 


40 

/ 


.98843 


1.01170 


20 


60 
/ 


1.00000 


1.00000 







Cotang 1 Tang 


/ 


Cotang 


Tang 


/ 


Cotang 


Tang 


/ 




45° 


45° 


45° 





TABLES. 



669 




HNK^O ©**•»©© «**©<»© « ^ © OC © ^X«!-CO 

h HHrt-iN e»NN«eo ec^^^'o 



55 

w 

a 

< 

Pi 
o 



w 

H 
H 



o 



w 
p 

> 

o 

as 



O 



»3 

00 
O 


co in m co 


NO M CO N 


Mntoo 01 


OnnO 


O "INNS 


vo w in r-~ 

11 O Pi N co 


Pt CO U1\0 t-» 
CO CO CO CO CO 


O - co ->f m 
co -f f ■* ->f 


r-co O T 
■<f -<r -t -^- m 


pi co -f inNO 
in in in in in 


© 
M 

O 


m co co N M 

NO CONO 
N pi co co co 


in co •*• On 

OO O N CO if 

ro •* •* •*- •*■ 


M N O NO 


PI CO ■* -<f PI 


00 co m N n 


t-. On p" Pt -<f 

xf -f m in m 


inNO r^co on 
m m in m 10 


c* c tj- in 

NO NO NO NO NO 


*0 
* 


OnnO ■*■<*■■* 


O w w 00 Tf 


h w co m 


O <-> PI CO CO 


moo h 


in coco n 10 


00 P) CO m 

Tf m m in m 


00 m -f in 
inNO no no no 


NO CO O « 
NO NO NO t^ (^ 


co Tf in r-.oo 
N c~ r^ n n 


15 
« 

© 


NO NO "^-00 


m 00 in 


co r-» OnnO ^f 


f"« n PI PI 


>n on N co 


00 -f CO w 

co co t t>- m 


■<f NO On N 

in 10 itjno NO 




TNO t^CO On 

t^ t-~ t^ t^ r~ 


m pi c^ invo 
CO 00 00 00 CO 


O 

© 


insoo n O 


On -f r>. N in 


in co m On 


CONO N CO 


H N CO *f NO 


co -*f m in*o 


NO NO NO NO t~H 


Tf r-> On m in 
n rr-co 00 


■* in r^oo On 

00 co 00 co 00 


H N ♦ >C NO 

On On On On On 


O 


VO w -^-oo CO 


■*• CO On N O 


■«f pi inNO •*• 


-*oo M 


H CO CONO ON 


-fNO COCO N 

n- io\o no c-« 


NO On " •*■ NO 

r-» r-~oo 00 00 


0> Pt -<fNC CO 
CO On On On ON 


O » w *m 
C 


r-^co M fi 
O *-* H M 


10 

H 
O 


pi co -<f "-• 


CO NO O H 


nnn mm 


O -<foo P) 


CO NO O PI 


cono 10 m 
mvo rvco 00 


O0 N * M3> 
00 ON ON On ON 


N in co pi 

O O IH M 


•*• "INC 00 


m co ^- in N 

PI PI PI Pi N 








M 

H 
O 


Pi pi pi N 


co t-sNO >n 


nj-SN MCI 


Ccn On CO r^. On 


00 M-co 


"1 O no h 

NO CO OS On O 

M 


moo « ■"f no 

O O HI 1- *4 


O CONO CO O 

« « n n m 


pt co m>c r^ 

CO CO CO CO CO 


O »-• co ■*}■ NO 










H 
O 


r^ - 00 O N 


N^«1H ■+ 


ION* NOO 


t^ cooo hi co 


f co ■*• N 


PI O00 NO w 
N0O ON « 


in on pi in r^ 

<-> 1-1 Pi (M PI 


M "J-SO>M 
CO CO '"O CO •^ 1 


c u- nc co 

■*■ -r -f f f 


>- ro inNO N 
in in m lti in 




H 
H 
O 


N 00 


CO On m co in 


NO OnOO P) •"f 


co O moo pi 


CO PI 00 CONO 


PI 1- 00 CO 

CO in M Pi 


00 h in cn 

pi CO CO CO ^f 


rf r-~ co 10 
Tf t m m m 


r-. On ►» CO 

in inNc no no 


in c^co « 




O 
O 


CO M CO IOVO 


co ->f 00 t>. M 


•^-0 inNO 


m pi 00 pi -<f 


l-~NO COOO w 


CO CO CO Pi O0 
Oi 1-1 Pi CO CO 


co n covo 
■^* Tf in in in 


O -t SOh 

NO NO NO NO f^ 


CO m.NO CO On 
t-^ r^ t-~ r^ ic~ 


1-. m "VC 00 
OC CO CO 00 00 




© 

© 
O 


CO lOCO CO00 


ON M noo ■*• 


0>0 NM * 


CiO tsO co 


t-> r-. Tf co 


CO O- •-• NO 
N *10-0 


M NO On N in 
NO NO no r» t-» 


O cono On m 
00 00 CO CO On 


co inio CO On 

On C' O O O 


m r<~ m r^oo 
0O 
pi ei pi pi pi 














ihnmtjiio «^ao«© c?^©x© R**flB>ooo ^or. «c© 
h T4T4i-tT4ct ei « s? oi m eo vi ^ ^ ta 



670 



SURVEYING. 



X 

I— I 

w 
-1 

PQ 
< 



o 

w 

P4 

O 
<! 

•5 
O 

H 



en W 

pi w 

<j . 

I* rt 

O •= 

b, ° 



Q 
Z 
O 
U 



m 

O 

U 

fa 
O 

H 

W 
W 



O 



u 



Cubic Feet 

per 

Second. 


HNMHUSCiSOftOHOUCOOOOOiflOvSOOOOOOOOOOO 
HTHeiNC0M'*^»0©J>00C5©WL'jr-OO©i0©©©©©©© 

H 


w 

© 
© 

H 

« 
W 

J 
< 


© 

©* 
1-1 


'*lC©^*i , • On m ei ro Th ir>NO vo t^oo O m n -<J-no tvoo n roiot^O rouiN N 


© 
© 


«o\oi^i-»x©N ni/, ^ ( o ^00 m n ci ■^t-vo 00 (J m Tf-vo o> •* «^ irnoN 





m^o t^oo 


35HM^©^°° ononO m m ■>*■ mvo 00 N Tj-t^ONd Tj-t^H *no\n 


rir(HH , h m m o N N n N a m rr ) pr>rororo^--*'^-ioiom iono 


© 


in t—oo 00 


SK'JIlSBMOOO M r«"> ■«»■"■> t^ f~ cm -<»-no on h •«->© <*■ t-» O o "■> 


HHHHHHfi4S<t N N N N N N mmncoco^^NHnui in\o no O 


© 


\o t~.oo cyvOc^iij^aogsQ^g^jfjj^^^. on n m-no 00 n "-> t-^ ->*-oo h u-> t-~ 




\0 NOO 


m© r* cs © h c? co "*©**©© h^< ^ °° *- tx n t-. h inoo h ■<*■ 

M h HHHN«'««n'n'n'n M «W co m 4 4 4- "^ ^no no' no' t^-r; 


© 
H 


O 00 O 

M 


M . 1" ^ ?0 NMTf(«5©X©H«^©©e»'*iN« H 1" t ^ f,vo . ° 1"^° 
M M H H NN««NNNei»MMMJ5r|('f'T|i 10 u 1 , to no n^ ^ t^ t-00 




t^OO M 

H M 


N tooo O h ro Tt-ja «J-.05HeO^©©H^©©Tj*?N» O mo *oo m in 

m h « et m « N N ' N ' N ' NMM 'e ' M eo,*' ) *Tjiic'ww' vovo ^f-^ 0000 


© 

JO 


M M 




© 

© 


OO N I' 


•*oo w mmtsovo - mwso m N«CC5«W©T$<a0HO0MXeOf-H 


H 


00 m m^ifiammNOiH m ifinso n -*vo rovo ©^©Mi^^©!©©!*}© 


HrtH 


H 


omh ro mvo "*-no on m m -*-<o t~- n -*"0 00 n no on moo to ?^ h X »0 © © H »Q 


M M M 


m n n cs P) conroforo^Tt-Tj--<*--<i-ioin iono vo ^.f.aeftrfl5©©H»H 

rHHHH 


© 


On N -<fNO 

M M M 


t^<NNO Onh PimSONO ro >noo N P-. w Tt-00 f> On f>00 lfl J* X ^ © ^1 
m e! pj ei rofOforoco-^-^-^ri-iriin io\o «« n t^oo 00 -j" q q ,' — ' „' 


u 

:s 
u 


u a 
v 
0.0 

CO 


rt«Wi«58«50i90WOiflOOOOeC»OWOOOSOOOeOOO 

HrtNNWM^^lO^i^OOCiCNJOr-OiOCO ©©©©©©© 

t-li-lfHHNNMM^>3©i'0©©© 

H 



TABLES. 



67I 



TABLE X. 
Volume of the Prismoidal Formula. 



OS 










Heights. 










Corrections 
























for tenths 
in height. 




1 


2 


3 


4 


5 


6 


7 


8 


9 


10 


1 





1 


1 


1 


2 


2 


2 


2 


3 


3 


.1 





2 


1 


1 


2 





3 


3 


4 


5 


6 


6 




2 





3 


1 


2 


3 


4 


5 


6 


6 


7 


8 


9 




3 





4 


1 


2 


4 


5 


6 


7 


9 


10 


11 


12 




4 


1 


5 


a 


—a 


—5 


—6 


—8 


—9 


—11 


—12 


—14 


—15 




5 


1 


6 


2 


4 


6 


7 


9 


11 


13 


15 


17 


19 




6 


1 


7 


2 


4 


6 


9 


11 


13 


15 


17 


19 


22 




7 


1 


8 


2 


5 


7 


10 


12 


1 5 


17 


20 


22 


25 




8 


1 


9 


3 


6 


8 


11 


14 


17 


19 


22 


25 


28 




Q 


1 


10 


3 


6 


9 


12 


15 


19 


22 


25 


28 


31 




11 


3 


7 


10 


14 


17 


20 


24 


27 


31 


34 


.1 





12 


4 


7 


11 


15 


19 


22 


26 


30 


33 


37 




2 


1 


13 


4 


8 


12 


16 


20 


24 


28 


32 


36 


40 




3 


1 


14 


4 


9 


13 


17 


22 


26 


30 


35 


39 


43 




4 


2 


15 


—5 


—9 


—14 


—19 


—23 


—28 


—32 


—37 


—42 


—46 




5 


2 


Hi 


5 


10 


15 


20 


25 


30 


35 


40 


44 


49 




6 


3 


17 


5 


10 


16. 


21 


26 


31 


37 


42 


47 


52 




7 


3 


18 


6 


11 


17 


22 


28 


33 


39 


44 


50 


56 




8 


4 


19 


6 


£2 


18 


23 


29 


35 


41 


47 


53 


59 




q 


4 


20 


6 


12 


19 


25 


31 


37 


43 


49 


56 


62 




21 


6 


13 


19 


26 


32 


39 


45 


52 


58 


65 


.t 


1 


22 


1 


14 


20 


27 


34 


41 


48 


54 


61 


68 




2 


2 


23 


7 


14 


21 


28 


35 


43 


50 


57 


64 


71 




3 


2 


24 


7 


15 


22 


30 


37 


44 


52 


59 


67 


74 




4 


3 


25 


-8 


— 15 


—23 


-31 


—39 


—46 


—54 


—62 


—69 


—77 




5 


4 


26 


8 


16 


24 


32 


40 


48 


56 


64 


72 


80 




6 


5 


27 


8 


17 


25 


33 


42 


50 


58 


67 


75 


83 




7 


5 


28 


9 


17 


26 


35 


43 


52 


60 


69 


78 


86 




8 


6 


29 


9 


18 


27 


36 


45 


54 


63 


72 


81 


90 







7 


30 


9 


19 


28 


37 


46 


56 


65 


74 


83 


93 




31 


10 


19 


29 


38 


48 


57 


67 


77 


86 


96 


.1 


1 


32 


10 


20 


30 


40 


49 


' 59 


69 


79 


89 


99 




2 


2 


33 


10 


20 


31 


41 


51 


61 


71 


81 


92 


102 




3 


3 


34 


10 


21 


31 


42 


52 


63 


73 


84 


94 


105 




4 


4 


35 


—11 


—22 


—32 


—43 


—54 


—65 


—76 


—86 


-97 


—108 




5 


5 


36 


11 


22 


33 


44 


56 


67 


78 


89 


100 


111 




6 


6 


37 


11 


23 


34 


46 


57 


69 


80 


91 


103 


114 




7 


8 


38 


12 


23 


35 


47 


59 


70 


82 


94 


106 


117 




8 


9 


39 


12 


24 


36 


48 


60 


72 


84 


96 


108 


120 




q 


10 


40 


12 


25 


37 


49 


62 


74 


86 


99 


111 


123 






41 


13 


25 


38 


51 


63 


76 


89 


101 


114 


127 


.1 


1 


42 


13 


26 


39 


52 


65 


78 


91 


104 


117 


130 




2 


3 


43 


13 


27 


40 


53 


66 


80 


93 


106 


119 


133 




3 


4 


44 


14 


27 


41 


54 


68 


81 


95 


109 


122 


136 




4 


6 


45 


—14 


-28 


—42 


—56 


—69 


-83 


—97 


—111 


—125 


—139 




5 


7 


46 


14 


28 


43 


57 


71 


85 


99 


114 


128 


142 




6 


8 


47 


15 


29 


44 


58 


73 


87 


102 


116 


131 


145 




7 


10 


48 


15 


30 


44 


59 


74 


89 


104 


119 


133 


148 




8 


11 


49 


15 


30 


45 


60 


76 


91 


106 


121 


136 


151 




q 


13 


50 


15 


31 


46 


62 


77 


93 


108 


123 


139 


154 






1 


2 


3 


4 


5 


6 


7 


8 


9 


10 


.1 


.2 


•3 


• 4 


•5 


.6 


• 7 


.8 


•9 


Corrections for 











1 


1 


1 


1 


1 


1 


tenth 


s 1 


1 w 


dth. 



6; 2 



SUH VE YING. 



TABLE X. — Continued. 
Volume of the Prismoidal Formula. 



C/5 










Heights. 








C< 


nrrections 


•a 




















f 


Dr tenths 




















1 


% 


1 


2 


3 


4 


5 


6 


7 


8 


9 


10 i! 


i height. 


51 


16 


31 


47 


63 


79 


94 


110 


126 


142 


157 


i 


2 


52 


16 


32 


48 


64 


80 


96 


112 


128 


144 


160 


2 


3 


53 


16 


33 


49 


65 


82 


98 


115 


131 


147 


163 


3 


5 


54 


17 


33 


50 


67 


83 


100 


117 


133 


150 


167 


4 


7 


55 


—17 


-34 


—51 


—68 


—85 


—102 


—119 


—136 


—153 


—170 


5 


8 


56 


17 


35 


52 


69 


86 


104 


121 


138 


156 


173 


6 


10 


57 


18 


35 


53 


70 


88 


106 


123 


141 


158 


176 


7 


12 


58 


18 


36 


54 


72 


90 


107 


125 


143 


161 


179 


8 


14 


59 


18 


36 


55 


73 


91 


109 


127 


146 


164 


182 


9 


15 


60 


19 


37 


50 


74 


93 


111 


130 


148 


167 


185 




61 


19 


38 


56 


75 


94 


113 


132 


151 


169 


188 


i 


o 


6'2 


19 


38 


57 


77 


96 


115 


134 


153 


172 


191 


2 


4 


63 


19 


39 


58 


78 


97 


117 


136 


156 


175 


194 


3 


6 


61 


20 


40 


59 


79 


99 


119 


138 


158 


178 


197 


4 


8 


65 


— 20 


—40 


—60 


—80 


—100 


—120 


—140 


—160 


—181 


—201 


5 


10 


66 


20 


41 


61 


81 


102 


122 


143 


163 


183 


204 


6 


12 


67 


21 


41 


62 


83 


103 


124 


145 


165 


186 


207 


7 


14 


68 


21 


42 


63 


84 


105 


126 


147 


168 


189 


210 


8 


16 


69 


21 


43 


64 


85 


100 


128 


149 


170 


192 


213 


9 


18 


70 


2.2 


43 


65 


86 


108 


130 


151 


173 


194 


216 




71 


22 


44 


66 


88 


100 


131 


153 


175 


197 


219 


T 


2 


72 


22 


44 


67 


89 


111 


133 


156 


178 


200 


222 


2 


5 


73 


23 


45 


68 


90 


113 


135 


158 


180 


203 


225 


3 


7 


71 


23 


46 


69 


91 


114 


137 


160 


183 


206 


228 


4 


9 


75 


03 


—46 


—69 


—93 


—116 


—139 


—162 


-185 


—208 


—231 


5 


12 


76 


23 


47 


70 


94 


117 


141 


164 


188 


211 


235 


6 


14 


77 


24 


48 


71 


95 


119 


1-13 


166 


190 


214 


238 


7 


16 


78 


24 


48 


72 


96 


120 


144 


169 


193 


217 


241 


8 


19 


79 


24 


49 


73 


98 


122 


146 


171 


195 


219 


244 


9 


21 


80 


25 


49 


74 


99 


123 


148 


173 


198 


222 


247 




81 


25 


50 


75 


100 


125 


150 


175 


200 


225 


250 


i 


3 


82 


25 


51 


76 


101 


127 


152 


177 


202 


228 


253 


2 


5 


83 


26 


51 


77 


102 


128 


154 


179 


205 


231 


256 


3 


8 


81 


26 


52 


78 


104 


130 


156 


181 


207 


233 


259 


4 


10 


85 


—26 


—52 


—79 


—105 


—131 


—157 


— 1S4 


—210 


—236 


—262 


5 


13 


86 


27 


53 


80 


106 


133 


159 


186 


212 


239 


265 


6 


16 


'87 


27 


54 


81 


107 


134 


161 


188 


215 


242 


269 


7 


18 


88 


27 


54 


81 


109 


136 


163 


190 


217 


244 


272 


8 


21 


89 


27 


55 


82 


110 


137 


165 


192 


220 


247 


275 


9 


24 


90 


28 


56 


83 


111 


139 


167 


194 


222 


250 


278 




91 


28 


56 


84 


112 


140 


169 


197 


225 


253 


281 


i 


3 


92 


28 


57 


85 


114 


142 


170 


199 


227 


256 


284 


2 


6 


93 


29 


57 


86 


115 


144 


172 


201 


230 


258 


287 


3 


9 


91 


29 


58 


87 


116 


145 


174 


203 


232 


261 


290 


4 


12 


95 


-29 


—59 


-88 


—117 


—147 


—176 


—205 


—235 


—264 . 


—293 


5 


15 


96 


30 


59 


89 


119 


148 


178 


207 


237 


267 


296 


6 


18 


97 


30 


60 


90 


120 


150 


180 


210 


240 


269 


299 


7 


21 


98 


30 


60 


91 


121 


151 


181 


212 


242 


272 


302 


8 


23 


99 


31 


61 


92 


122 


153 


183 


214 


244 


275 


306 


9 


26 


100 


31 


62 


93 


123 


154 


185 


216 


247 


278 


309 






1 


2 


3 


4 


5 


•6 


7 


8 


9 


10 


. i 


.2 


•3 


•4 


•5 


.6 


•7 


.8 


•9 


























Correct 
tenths ii 


ons for 
l width. 






! 








1 


1 


1 


1 


1 


1 



TABLES. 



673 



TABLE X. — Continued. 
Volume of the Prismoidal Formula. 



tfl 

ja 










Heights. 










Corrections 
























for tenths 
in height. 


"0 


11 


12 


13 


14 


15 


16 


17 


18 


19 


20 


1 


3 


4 


4 


4 


5 


5 


5 


6 


6 


6 


.1 





2 


7 


7 


8 


9 


9 


10 


10 


11 


12 


12 




2 





3 


10 


11 


12 


13 


14 


15 


16 


17 


18 


19 




3 





4 


14 


15 


16 


17 


19 


20 


21 


22 


23 


25 




4 


1 


5 


—17 


—19 


—20 


—22 


—23 


—25 


—26 


—28 


—29 


-31 




5 


1 


6 


20 


22 


24 


26 


28 


30 


31 


33 


35 


37 




6 


1 


7 


24 


26 


28 


30 


32 


35 


37 


39 


41 


43 




7 


1 


8 


27 


30 


32 


35 


37 


40 


42 


44 


47 


49 




8 


1 


9 


31 


33 


36 


39 


42 


44 


47 


50 


53 


56 







1 


10 


34 


37 


40 


43 


46 


49 


52 


56 


59 


62 




11 


37 


41 


44 


48 


51 


54 


58 


61 


65 


68 


. 1 





12 


41 


44 


48 


52 


56 


59 


63 


67 


70 


74 




2 


1 


13 


44 


48 


52 


56 


60 


64 


68 


72 


76 


80 




3 


1 


14 


48 


52 


56 


60 


65 


69 


73 


78 


82 


86 




4 


2 


IS 


—51 


—56 


—60 


—65 


—69 


—74 


—79 


-83 


—88 


—93 




5 


2 


16 


54 


59 


64 


69 


74 


79 


84 


89 


94 


99 




6 


3 


17 


58 


63 


68 


73 


79 


84 


89 


94 


100 


105 




7 


3 


18 


61 


67 


72 


78 


83 


89 


94 


100 


106 


111 




8 


4 


19 


65 


70 


76 


82 


88 


94 


100 


106 


111 


117 




Q 


4 


20 


68 


74 


80 


86 


93 


99 


105 


111 


117 


123 




21 


71 


78 


84 


91 


97 


104 


110 


117 


123 


130 


.1 


1 


22 


75 


81 


88 


95 


102 


109 


115 


122 


129 


136 




2 


2 


23 


78 


85 


92 


99 


106 


114 


121 


128 


135 


142 




3 


2 


24 


81 


89 ■ 


96 


104 


111 


119 


126 


133 


141 


148 




4 


3 


25 


—85 


—93 


—100 


—108 


—116 


—123 


—131 


—139 


—147 


—154 




5 


4 


26 


88 


96 


104 


112 


120 


128 


136 


144 


152 


160 




6 


5 


27 


92 


100 


108 


117 


125 


133 


142 


150 


158 


167 




7 


5 


28 


95 


104 


112 


121 


130 


138 


147 


156 


164 


173 




8 


6 


29 


98 


107 


116 


125 


134 


143 


152 


161 


170 


179 




Q 


7 


30 


102 


111 


120 


130 


139 


148 


157 


167 


176 


185 




31 


105 


115 


*124 


134 


144 


153 


163 


172 


1^2 


191 


.1 


1 


32 


109 


119 


128 


138 


148 


158 


168 


178 


188 


198 




2 


2 


33 


112 


122 


132 


143 


153 


163 


173 


183 


194 


204 




2 


3 


34 


115 


126 


136 


147 


157 


168 


118 


189 


109 


210 




4 


4 


35 


—119 


—130 


—140 


—151 


—162 


—173 


—184 


—194 


-205 


-216 




5 


5 


36 


122 


133 


144 


156 


167 


178 


189 


200 


2J1 


222 




6 


6 


37 


126 


137 


148 


160 


171 


183 


194 


206 


217 


228 




7 


8 


38 


129 


141 


152 


164 


176 


188 


199 


211 


223 


235 




8 


9 


39 


132 


144 


156 


169 


181 


193 


205 


217 


229 


241 




q 


10 


40 


136 


148 


160 


173 


185 


198 


210 


222 


235 


247 




41 


139 


152 


165 


177 


190 


202 


215 


228 


240 


253 


. 1 


1 


42 


143 


156 


169 


181 


194 


207 


220 


233 


246 


259 




2 


3 


43 


146 


159 


173 


186 


199 


212 


226 


239 


252 


265 




3 


4 


44 


149 


163 


177 


190 


204 


217 


231 


244 


258 


272 




4 


6 


45 


—153 


—167 


-181 


—104 


—208 


—222 


—236 


—250 


—264 


—278 




5 


7 


46 


156 


170 


185 


199 


213 


227 


241 


256 


270 


284 




6 


8 


47 


160 


174 


189 


203 


218 


232 


247 


261 


276 


290 




7 


10 


48 


163 


178 


193 


2(17 


222 


237 


252 


267 


281 


296 




8 


11 


49 


166 


181 


197 


212 


227 


242 


257 


272 


287 


302 




Q 


13 


50 


170 


185 


201 


216 


231 


247 


262 


278 


293 


309 






11 


12 


13 


14 


15 


16 


17 


18 


19 


20 


.1 


.2 


•3 


•4 


•5 


.6 


■7 


.8 


•9 


Corrections for 





1 


1 


2 


2 


3 


3 


4 


4 


tenth 


s 1 


n w 


dth. 



674 



SURVEYING. 



TABLE X. — Continued. 
Volume of the Prismoidal Formula. 



en 










Heights. 








Cc 


>rrections 


■*-* 




















f ( 


)1" tC*"**"^" 10 




11 


12 


13 


14 


15 


16 


17 


18 


19 


20 h 


i he 


ght. 


51 


173 


189 


205 


220 


236 


252 


268 


283 


299 


315 


i 


2 


52 


177 


193 


209 


225 


241 


257 


273 


289 


305 


321 


2 


3 


53 


180 


196 


213 


229 


245 


262 


278 


294 


311 


327 


3 


5 


54 


183 


200 


217 


233 


250 


267 


283 


300 


317 


333 


4 


7 


55 


—187 


—204 


—221 


—238 


—255 


—272 


—289 


—306 


—323 


—340 


5 


8 


56 


190 


207 


225 


242 


259 


277 


294 


311 


328 


346 


6 


10 


57 


194 


211 


229 


246 


264 


281 


299 


317 


334 


352 


7 


12 


58 


197 


215 


233 


251 


269 


286 


304 


322 


340 


358 


8 


14 


59 


200 


219 


237 


255 


273 


291 


310 


328 


346 


364 


9 


15 


60 


204 


222 


241 


259 


278 


296 


315 


333 


352 


370 






61 


207 


226 


245 


264 


282 


301 


320 


339 


358 


377 


i 


2 


62 


210 


230 


249 


268 


287 


306 


325 


344 


364 


383 


2 


4 


63 


214 


233 


253 


272 


292 


311 


331 


350 


369 


389 


3 


6 


61 


217 


237 


257 


277 


296 


316 


336 


356 


375 


395 


4 


8 


65 


-221 


-241 


—261 


—281 


—301 


—321 


—341 


—361 


—381 


-401 


5 


10 


66 


224 


244 


265 


285 


306 


326 


346 


367 


387 


407 


6 


12 


67 


227 


248 


269 


290 


310 


331 


352 


372 


393 


414 


7 


14 


68 


231 


252 


273 


294 


315 


336 


357 


378 


399 


420 


8 


16 


69 


234 


256 


277 


298 


319 


311 


362 


383 


405 


426 


9 


18 


70 


238 


259 


281 


302 


324 


346 


367 


389 


410 


432 






71 


241 


263 


285 


307 


329 


351 


373 


394 


416 


438 


i 


2 


72 


244 


267 


289 


311 


333 


356 


378 


400 


422 


444 


2 


5 


73 


248 


270 


293 


315 


338 


360 


383 


406 


428 


451 


3 


7 


74 


251 


274 


297 


320 


343 


365 


388 


411 


434 


457 


4 


9 


75 


—255 


—278 


-301 


—324 


—347 


—370 


—394 


—417 


-440 


—463 


5 


12 


76 


258 


281 


305 


328 


352 


375 


399 


422 


446 


469 


6 


14 


77 


261 


285 


309 


333 


356 


380 


404 


428 


452 


475 


• 7 


16 


78 


2G5 


289 


313 


337 


361 


385 


409 


433 


457 


481 


.8 


19 


79 


268 


293 


317 


341 


366 


390 


415 


439 


463 


488 


•9 


21 


80 


272 


296 


321 


346 


370 


395 


420 


444 


469 


494 






81 


275 


300 


325 


350 


375 


400 


425 


450 


475 


500 


.i 


3 


82 


278 


304 


329 


354 


380 


405 


430 


456 


481 


506 


.2 


5 


8:; 


282 


307 


333 


359 


384 


410 


435 


461 


487 


512 


•3 


8 


84 


285 


311 


337 


363 


389 


415 


441 


467 


493 


519 


•4 


10 


85 


—289 


—315 


—341 


-367 


—394 


—420 


—446 


-472 


—498 


-525 


•5 


13 


86 


292 


319 


345 


372 


398 


425 


451 


478 


504 


531 


.6 


16 


87 


295 


322 


349 


376 


403 


430 


456 


483 


510 


537 


7 


18 


88 


299 


326 


353 


380 


407 


435 


462 


489 


516 


543 


.8 


21 


89 


303 


330 


357 


385 


412 


440 


467 


494 


522 


549 


•9 


24 


90 


306 


333 


361 


389 


417 


444 


472 


500 


528 


556 






91 


309 


337 


365 


393 


421 ' 


449 


477 


506 


534 


562 


i 


8 


92 


312 


341 


369 


398 


426 


454 


483 


511 


540 


568 


2 


6 


93 


316 


344 


373 


402 


431 


459 


488 


517 


545 


574 


3 


9 


94 


319 


348 


377 


406 


435 


464 


493 


522 


551 


580 


4 


12 


95 


—323 


—352 


-381 


—410 


—440 


-469 


—498 


—528 


—557 


—586 


5 


15 


96 


326 


356 


385 


415 


444 


474 


504 


533 


563 


593 


6 


18 


97 


329 


359 


389 


419 


449 


479 


509 


539 


569 


599 


7 


21 


98 


333 


363 


393 


423 


454 


484 


514 


544 


575 


605 


8 


23 


99 


336 


367 


397 


428 


458 


489 


519 


550 


581 


611 


9 


26 


100 


340 


370 


401 


432 


463 


494 


525 


556 


586 


617 
~20 








11 


12 


13 


14 


15 


16 


17 


18 


19 


.i 


.2 


•3 


•4 


•5 


.6 


•7 


.8 


•9 


Correct 
tenths i 


ions 
i wi 


for 
dth. 





1 


1 


2 


2 


3 


3 


4 


i 4 



TABLES. 



675 



TABLE X. — Continued. 
Volume of the Prismoidal Formula. 



ID 


Heights. 


1 

Corrections 
for tenths 


*o 






















5 


21 


22 


23 


24 


25 


26 


27 


28 


29 


30 


in height. 


1 


6 


7 


7 


7 


8 


8 


8 


9 


9 


9 


.1 





2 


13 


14 


14 


15 


15 


16 


17 


17 


18 


19 




2 





3 


19 


20 


21 


22 


23 


24 


25 


26 


27 


28 




3 





4 


26 


27 


28 


30 


31 


32 


33 


35 


36 


37 




4 


1 


5 


—32 


—34 


—35 


—37 


—39 


—40 


—42 


—43 


—45 


—46 




5 


1 


6 


39 


41 


43 


44 


46 


48 


50 


52 


54 


56 




6 


1 


7 


45 


48 


50 


52 


54 


56 


58 


60 


63 


65 




7 


1 


8 


52 


54 


57 


59 


62 


64 


67 


69 


72 


74 




8 


1 


9 


58 


61 


64 


67 


69 


72 


75 


78 


81 


83 







1 


10 


65 


68 


71 


74 


77 


80 


83 


86 


90 


93 




11 


71 


75 


78 


81 


85 


88 


92 


95 


98 


102 


.1 





12 


78 


81 


85 


89 


93 


96 


100 


104 


107 


111 




2 


1 


13 


84 


88 


92 


96 


100 


114 


108 


112 


116 


120 




3 


1 


14 


91 


95 


99 


104 


108 


112 


117 


121 


125 


130 




4 


2 


15 


-97 


—102 


—106 


—111 


—116 


—120 


—125 


—130 


—134 


—139 




5 


2 


16 


104 


109 


114 


119 


123 


128 


133 


138 


143 


148 




6 


3 


17 


110 


115 


121 


126 


131 


136 


142 


147 


152 


157 




7 


3 


18 


117 


122 


128 


133 


139 


144 


150 


156 


161 


167 




8 


4 


19 


123 


129 


135 


141 


147 


152 


158 


164 


170 


176 




Q 


4 


20 


130 


136 


142 


148 


154 


160 


167 


173 


179 


185 






21 


136 


143 


149 


156 


162 


169 


175 


181 


188 


194 


.1 


1 


22 


143 


149 


156 


163 


170 


177 


183 


190 


197 


204 




2 


2 


23 


149 


156 


163 


170 


177 


185 


192 


199 


206 


213 




-. 


2 


24 


156 


163 


170 


178 


185 


193 


200 


207 


215 


222 




4 


3 


25 


—1(52 


—170 


—177 


-185 


—193 


—201 


-208 


—216 


—224 


—231 




5 


4 


26 


169 


177 


185 


193 


201 


209 


217 


225 


233 


241 




6 


5 


27 


175 


183 


192 


200 


208 


217 


225 


233 


242 


250 




7 


5 


28 


181 


190 


199 


207 


216 


225 


233 


242 


251 


259 




8 


6 


29 


188 


197 


206 


215 


224 


233 


242 


251 


260 


269 




Q 


7 


30 


194 


204 


213 


222 


231 


241 


250 


259 


269 


278 




31 


201 


210 


220 


230 


239 


249 


258 


268 


277 


287 


.1 


1 


32 


207 


217 


227 


237 


247 


257 


267 


277 


286 


296 




2 


2 


33 


214 


224 


234 


244 


255 


265 


275 


285 


295 


306 




3 


3 


34 


220 


231 


241 


252 


262 


273 


283 


294 


304 


315 




4 


4 


35 


—227 


—238 


—248 


—259 


—270 


—281 


—292 


—302 


—313 


-324 




5 


5 


36 


233 


244 


256 


267 


278 


289 


300 


311 


322 


333 




6 


6 


37 


240 


251 


263 


274 


285 


297 


308 


320 


331 


343 




7 


8 


38 


246 


258 


270 


281 


293 


305 


317 


328 


3-10 


352 




8 


9 


39 


253 


265 


277 


289 


an 


313 


325 


337 


349 


361 




Q 


10 


40 


259 


272 


284 


296 


309 


321 


333 


346 


358 


370 




41 


266 


278 


291 


304 


316 


329 


342 


354 


367 


380 


.1 


1 


42 


272 


285 


298 


311 


324 


337 


350 


363 


376 


389 




2 


3 


43 


279 


292 


305 


319 


332 


345 


358 


372 


385 


398 




3 


4 


44 


285 


299 


312 


326 


340 


353 


367 


380 


394 


407 




4 


6 


45 


—292 


—306 


—319 


—333 


—347 


—361 


—375 


-389 


—403 


—417 




5 


7 


46 


298 


312 


327 


341 


355 


369 


383 


398 


412 


426 




6 


8 


47 


305 


319 


334 


348 


363 


377 


392 


406 


421 


435, 




7 


10 


48 


311 


326 


341 


356 


370 


385 


400 


415 


430 


444 




S 


11 


49 


318 


333 


348 


363 


378 


393 


408 


423 


439 


454 







13 


50 


324 


340 


355 


370 


386 


401 


417 


432 


448 


463 






21 


22 


23 


24 


25 


26 


27 


28 


29 


30 


.1 


.2 


•3 


•4 


•5 


.6 


•7 


.8 


•9 


Corrections for 


1 


2 


2 


3 


4 


5 


5 


6 


7 


tenth 


s i 


1 WI 


dth. 



6y6 



SURVEYING. 



TABLE X. — Continued. 
Volume of thk Prismoidal Formula. 



10 










Heights. 








Corrections 






















— — — ^— Tf"n* tpnthc 


* m 


21 


22 


23 


24 


25 


26 


21 


28 


29 


1\J1 LCll Lllo 

30 in height. 


51 


331 


346 


362 


378 


394 


409 


425 


441 


456 


472 


i 


2 


52 


337 


353 


369 


385 


401 


417 


433 


449 


465 


481 


2 


3 


53 


344 


360 


376 


393 


409 


425 


442 


458 


474 


491 


3 


5 


54 


350 


367 


383 


400 


417 


433 


450 


467 


483 


500 


4 


7 


55 


-356 


-373 


-390 


-407 


—424 


—441 


-458 


—475 


-492 


-509 


5 


8 


56 


363 


380 


398 


415 


432 


449 


467 


484 


501 


519 


6 


10 


57 


369 


387 


405 


422 


440 


457 


475 


493 


510 


528 


7 


12 


58 


376 


394 


412 


430 


448 


465 


483 


501 


519 


537 


8 


14 


59 


382 


401 


419 


437 


455 


4:3 


492 


510 


528 


546 


9 


15 


60 


389 


407 


426 


444 


463 


481 


500 


519 


537 


556 




61 


395 


414 


433 


452 


471 


490 


508 


527 


546 


565 


i 


2 


62 


402 


421 


440 


459 


478 


438 


517 


536 


555 


574 


2 


4 


63 


408 


428 


447 


467 


486 


506 


525 


544 


564 


583 


3 


6 


61 


415 


435 


454 


474 


494 


514 


533 


553 


573 


593 


4 


8 


65 


—421 


—441 


—461 


—481 


—502 


—522 


—542 


-562 


—582 


—602 


5 


10 


66 


428 


448 


469 


489 


509 


530 


550 


570 


591 


611 


6 


12 


67 


431 


455 


476 


496 


517 


538 


558 


579 


600 


620 


7 


14 


68 


441 


462 


483 


504 


525 


546 


567 


588 


609 


630 


8 


36 


69 


447 


469 


490 


511 


532 


554 


575 


596 


618 


639 


9 


18 


70 


454 


475 


497 


519 


540 


562 


583 


605 


627 


648 




71 


460 


482 


504 


526 


548 


570 


592 


614 


635 


657 


i 


2 


72 


467 


489 


511 


533 


556 


578 


600 


622 


644 


667 


2 


5 


73 


473 


496 


518 


541 


563 


586 


608 


631 


653 


676 


3 


7 


74 


480 


502 


525 


548 


571 


594 


617 


640 


662 


685 


4 


9 


75 


—486 


—509 


—532 


—556 


—579 


-601 


—625 


—648 


—671 


—694 


5 


12 


76 


493 


516 


540 


563 


586 


610 


633 


657 


680 


704 


6 


14 


77 


499 


523 


547 


570 


594 


618 


642 


665 


689 


713 


7 


16 


78 


506 


530 


554 


578 


602 


626 


650 


674 


698 


722 


8 


19 


79 


512 


536 


561 


585 


610 


634 


658 


683 


707 


731 


9 


21 


80 


519 


543 


568 


593 


617 


642 


667 


691 


716 


741 




81 


525 


550 


575 


600 


625 


650 


675 


700 


725 


750 


.i 


3 


82 


531 


557 


582 


607 


633 


658 


683 


709 


734 


759 


.2 


5 


83 


538 


564 


589 


615 


640 


666 


692 


717 


743 


769 


•3 


8 


84 


544 


570 


596 


622 


648 


674 


700 


726 


752 


778 


•4 


10 


85 


—551 


—577 


—603 


-630 


-656 


—682 


—708 


—735 


—761 


—787 


•5 


13 


86 


557 


584 


610 


637 


664 


690 


717 


743 


770 


796 


.6 


16 


87 


564 


591 


618 


644 


671 


698 


725 


752 


779 


806 


•7 


18 


88 


570 


598 


625 


652 


679 


706 


733 


760 


788 


815 


.8 


21 


89 


577 


604 


632 


659 


687 


714 


742 


769 


797 


824 


•9 


24 


90 


583 


611 


639 


667 


694 


722 


750 


777 


806 


833 




91 


590 


618 


646 


674 


702 


730 


758 


786 


815 


843 


.i 


3 


92 


596 


625 


653 


681 


710 


738 


767 


795 


823 


852 


.2 


6 


93 


603 


631 


660 


689 


718 


746 


775 


804 


832 


861 


•3 


9 


94 


609 


638 


667 


696 


725 


754 


783 


812 


841 


870 


-4 


12 


95 


—616 


—645 


-674 


—704 


—733 


—762 


—792 


—821 


—850 


—880 


■5 


15 


96 


622 


652 


681 


711 


741 


770 


800 


830 


859 


889 


.6 


18 


97 


629 


659 


689 


719 


748 


778 


808 


838 


868 


898 


•7 


21 


90 


C35 


665 


696 


726 


756 


786 


817 


847 


877 


907 


.8 


23 


99 


642 


672 


703 


733 


764 


794 


825 


856 


886 


917 


•9 


26 


100 


G48 


679 


710 


741 


772 


802 


833 


864 


895 


926 






21 


22 


23 


24 


25 
•5 


26 


27 


28 


29 


30 


.i 


.2 


•3 


•4 
3 


.6 


•7 


.8 


•9 


Correc 
tenths i 


lions for 


1 


2 


2 


4 


5 


5 


6 


7 


n w 


ldth. 



TABLES. 



677 



TABLE X. — Continued. 
Volume of the Prismoidal Formula. 



.C 










Heights. 








Corrections 






















- . . io r fe,nt ' ,|!: ' 


ID 


31 


32 


33 


34 


35 


36 


37 


38 


39 


40 » 


_»1 L^MLIl.-> 

i height. 


1 


10 


10 


10 


10 


11 


11 


11 


12 


12 


12 


1 





2 


19 


20 


20 


21 


22 


22 


23 


23 


24 


25 


2 





3 


29 


30 


31 


31 


32 


33 


34 


35 


36 


37 


3 





4 


33 


40 


41 


42 


43 


44 


46 


47 


48 


49 


4 


1 


5 


—18 


-49 


—51 


—52 


—54 


—56 


—57 


-59 


—60 


—62 


5 


1 


6 


57 


59 


61 


63 


65 


67 


68 


70 


72 


74 


6 


1 


7 


67 


69 


71 


73 


76 


78 


80 


82 


84 


86 


7 


1 


8 


77 


79 


81 


84 


86 


89 


91 


94 


96 


97 


8 


1 


9 


86 


89 


92 


94 


97 


100 


103 


106 


108 


111 


9 


1 


10 


96 


99 


102 


105 


108 


111 


114 


117 


120 


123 




11 


105 


109 


112 


115 


119 


122 


126 


129 


132 


136 


1 





12 


115 


119 


122 


126 


130 


133 


137 


141 


144 


148 


2 


1 


13 


124 


128 


132 


136 


140 


144 


148 


152 


156 


160 


3 


1 


14 


134 


138 


143 


147 


151 


156 


160 


164 


169 


173 


4 


2 


15 


—144 


—148 


—153 


—157 


—162 


—167 


—171 


—176 


—181 


—185 


5 


2 


16 


153 


158 


163 


168 


173 


178 


183 


188 


193 


198 


6 


3 


17 


163 


168 


173 


178 


183 


189 


194 


199 


205 


210 


7 


3 


18 


172 


178 


183 


189 


194 


200 


206 


211 


217 


222 


8 


4 


19 


182 


188 


194 


199 


205 


211 


217 


223 


229 


235 


9 


4 


20 


191 


198 


204 


210 


216 


222 


228 


235 


241 


247 




21 


201 


207 


214 


220 


227 


233 


240 


246 


253 


259 


1 


1 


22 


210 


217 


224 


231 


238 


244 


251 


258 


265 


272 


2 


2 


23 


220 


227 


234 


241 


248 


256 


263 


270 


277 


284 


3 


2 


24 


230 


237 


244 


252 


259 


267 


274 


281 


289 


296 


4 


3 


25 


-239 


—247 


—255 


—262 


—270 


—278 


—285 


—293 


—301 


—309 


5 


4 


26 


249 


*57 


265 


273 


281 


289 


297 


305 


313 


321 


6 


5 


27 


258 


267 


275 


283 


292 


300 


308 


317 


325 


333 


7 


5 


28 


208 


277 


285 


294 


302 


311 


320 


328 


337 


346 


8 


6 


29 


277 


286 


295 


304 


313 


322 


331 


340 


349 


358 


9 


7 


30 


287 


296 


306 


315 


324 


333 


343 


352 


361 


370 




31 


297 


306 


316 


325 


335 


344 


354 


364 


373 


383 


.1 


1 


32 


306 


316 


326 


336 


346 


356 


365 


375 


385 


395 


2 





33 


316 


326 


336 


346 


356 


367 


377 


387 


397 


407 


3 


3 


31 


325 


336 


346 


357 


367 


378 


388 


399 


409 


420 


4 


4 


35 


-335 


—346 


—356 


—367 


—378 


—389 


—400 


—410 


—421 


—432 


5 


5 


36 


344 


356 


367 


378 


389 


400 


411 


422 


433 


444 


6 


6 


37 


354 


365 


377 


388 


400 


411 


423 


434 


445 


457 


7 


8 


38 


364 


375 


387 


399 


410 


422 


434 


446 


457 


469 


8 


9 


39 


373 


385 


397 


409 


421 


433 


445 


457 


469 


481 


9 


10 


40 


383 


395 


407 


420 


432 


444 


457 


469 


481 


494 




41 


392 


405 


413 


430 


443 


456 


468 


481 


494 


506 


1 


1 


42 


402 


415 


428 


441 


454 


467 


480 


403 


506 


519 


2 


3 


43 


411 


425 


438 


451 


465 


478 


491 


504 


518 


531 


3 


4 


44 


421 


435 


448 


462 


475 


489 


502 


516 


530 


543 


4 


6 


45 


—431 


—444 


—458 


—472 


—486 


—500 


—514 


—528 


— 542 


—556 


5 


7 


46 


440 


454 


469 


483 


497 


511 


525 


540 


554 


568 


6 


8 


47 


450 


464 


479 


493 


508 


522 


537 


551 


566 


580 


7 


10 


48 


459 


474 


489 


504 


519 


533 


548 


563 


578 


593 


8 


11 


49 


4(59 


484 


499 


514 


529 


544 


560 


575 


590 


605 


9 


13 


50 


478 


494 


509 


525 


540 


556 


571 


586 


602 


617 






31 


32 


33 


34 


35 


36 


37 


38 


39 


40 


. 1 


.2 


•3 


•4 


•5 


.6 


•7 


.8 


•9 


Correct 
tenths i 


ions for 


1 


2 


3 


4 


5 


6 


8 


9 


10 


1 w 


dth. 



678 



SURVEYING. 



TABLE ^.—Continued. 
Volume of the Prismoidal Formula. 



Cfl 










Heights. 








1 

! c 


orrections 


td 




















j for tenths 


S 


31 


32 


33 


34 


35 


36 


37 


38 


39 


40 in height. 


51 


488 


504 


519 


535 


551 


567 


582 


598 


614 


630 


.1 


2 


52 


498 


514 


530 


546 


562 


578 


594 


610 


626 


642 


.2 


3 


53 


507 


523 


540 


556 


573 


589 


605 


622 


638 


654 


.3 


5 


54 


517 


533 


55G 


567 


583 


600 


617 


633 


650 


667 


•4 


7 


55 


—526 


—543 


—560 


—577 


—594 


—611 


—628 


—645 


—662 


—679 


.5 


8 


56 


536 


553 


570 


588 


605 


622 


640 


657 


674 


691 


.6 


10 


57 


545 


563 


581 


598 


616 


633 


651 


669 


686 


704 


.7 


12 


58 


555 


573 


591 


609 


627 


644 


662 


680 


698 


716 


.8 


14 


59 


565 


583 


601 


619 


637 


656 


674 


692 


710 


728 


•9 


15 


60 


574 


593 


611 


630 


648 


667 


685 


704 


722 


741 




61 


584 


602 


621 


640 


659 


678 


697 


715 


734 


753 


.1 


2 


62 


593 


612 


631 


651 


670 


689 


708 


727 


746 


765 


.2 


4 


63 


603 


'622 


642 


661 


681 


700 


719 


739 


758 


778 


.3 


6 


61 


612 


632 


652 


672 


691 


711 


731 


751 


770 


790 


4 


8 


65 


—622 


—642 


—662 


— 6S2 


—702 


—722 


—742 


—762 


—782 


—802 


5 


10 


66 


631 


652 


672 


693 


713 


733 


754 


774 


794 


815 


6 


12 


67 


641 


662 


682 


703 


724 


744 


765 


786 


806 


827 


7 


14 


68 


651 


672 


693 


714 


735 


756 


777 


798 


819 


840 


8 


16 


69 


660 


681 


703 


724 


745 


767 


788 


809 


831 


852 


9 


18 


70 


670 


691 


713 


735 


756 


778 


799 


821 


843 


864 




71 


679 


701 


723 


745 


767 


789 


811 


833 


855 


877 


1 


2 


72 


689 


711 


733 


756 


778 


800 


822 


844 


867 


889 


2 


5 


73 


698 


721 


744 


766 


789 


811 


834 


856 


879 


901 


3 


7 


74 


708 


731 


754 


777 


799 


822 


845 


868 


891 


914 


4 


9 


75 


—718 


—741 


—764 


—787 


—810 


—833 


—856 


—880 


—903 


—926 


5 


12 


76 


727 


751 


774 


798 


821 


844 


868 


891 


915 


938 


6 


14 


77 


737 


760 


784 


808 


832 


856 


879 


903 


927 


951 


7 


16 


78 


746 


7^0 


794 


819 


843 


867 


891 


915 


939 


963 


8 


19 


79 


756 


780 


805 


829 


853 


878 


902 


927 


951 


975 


9 


21 


80 


765 


790 


815 


840 


864 


889 


914 


938 


963 


988 




81 


775 


800 


825 


850 


875 


900 


925 


950 


975 


1000 


1 


3 


82 


785 


810 


835 


860 


886 


911 


936 


962 


987 


1012 


2 


5 


83 


794 


820 


845 


871 


897 


922 


948 


973 


999 


1025 


3 


8 


84 


804 


830 


856 


881 


907 


933 


959 


' 985 


1011 


1037 


4 


10 


85 


—813 


—840 


—866 


—892 


—918 


—944 


—971 


—997 


—1023 


—1049 


5 


13 


86 


823 


849 


876 


902 


929 


956 


982 


1009 


1035 


1062 


6 


16 


87 


832 


859 


886 


913 


940 


967 


994 


1020 


1047 


1074 


7 


18 


88 


842 


869 


896 


923 


951 


978 


1005 


1032 


1059 


1086 


8 


21 


89 


852 


879 


906 


934 


961 


989 


1016 


1044 


1071 


1098 


9 


24 


90 


861 


889 


917 


944 


972 


1000 


1028 


1056 


1083 


1111 




91 


871 


899 


927 


955 


983 


1011 


1039 


1067 


1095 


1123 


1 


3 


92 


880 


909 


937 


965 


994 


1022 


1051 


1079 


1107 


1136 


2 


6 


93 


890 


919 


947 


976 


1005 


1033 


1062 


1091 


1119 


1148 


3 


9 


94 


899 


928 


957 


986 


1015 


1044 


1073 


1102 


1131 


1160 


4 


12 


95 


—909 


—938 


—968 


—997 


—1026 


—1056 


—1085 


—1114 


—1144 


—1173 


5 


15 


96 


919 


948 


978 


1007 


1037 


1067 


1096 


1126 


1156 


1185 


6 


18 


97 


928 


958 


988 


1018 


1048 


1078 


1108 


1138 


1168 


1198 


7 


21 


98 


938 


968 


998 


1028 


1059 


1089 


1119 


1149 


1180 


1210 


8 


23 


99 


947 


978 


1008 


1039 


1069 


1100 


1131 


1161 


1192 


1222 


9 


26 


100 


957 


988 


1019 


1049 


1080 


1111 


1142 


1173 


1204 


1235 






31 


32 


33 


34 


35 


36 


37 


38 


39 


40 


. 1 


.2 


•3 


•4 


•5 


.6 


•7 


.8 


•9 


Correct 
tenths ii 


ions for 


1 


2 


3 


4 


5 


6 


8 


9 


10 


1 Wl 


dth. 



TABLES. 



6/9 



TABLE X.— Continued. 
Volume of the Prismoidal Formula. 



j3 










Heights. 








C 


Directions 






















f 


or tenths 
n height. 


•a 


41 


42 


43 


44 


45 


46 


47 


48 


49 


50 * 


1 


13 


13 


13 


14 


14 


14 


15 


15 


15 


15 | 


i 





2 


25 


26 


27 


27 


28 


28 


29 


30 


30 


31 


2 





3 


38 


39 


40 


41 


42 


43 


44 


44 


45 


46 


3 





4 


51 


52 


53 


54 


56 


57 


58 


59 


60 


62 


4 


1 


5 


— G3 


—65 


—66 


—68 


—69 


—71 


—73 


—74 


—76 


—77 


5 


1 


6 


76 


78 


80 


81 


83 


85 


87 


89 


91 


93 


.6 


1 


7 


89 


91 


93 


95 


97 


99 


102 


104 


106 


108 


7 


1 


S 


101 


104 


106 


109 


111 


114 


116 


119 


121 


123 


8 


1 


9 


114 


117 


119 


1-22 


125 


128 


131 


133 


136 


139 


•9 


1 


10 


127 


130 


133 


136 


139 


142 


145 


148 


151 


154 




11 


139 


143 


146 


149 


153 


156 


160 


163 


166 


170 


i 





12 


152 


156 


159 


163 


167 


170 


174 


178 


181 


185 


2 


1 


13 


165 


169 


173 


177 


181 


185 


189 


193 


197 


201 


j 


1 


11 


177 


181 


186 


190 


194 


199 


203 


207 


212 


216 


4 


2 


15 


—190 


—194 


—199 


—204 


—208 


—213 


—218 


—222 


—227 


—231 


5 


2 


16 


203 


207 


212 


217 


222 


227 


232 


237 


242 


247 


6 


3 


17 


215 


220 


226 


231 


236 


241 


247 


252 


257 


262 


7 


3 


18 


228 


233 


239 


244 


250 


256 


261 


267 


272 


278 


8 


4 


19 


240 


246 


252 


258 


264 


270 


276 


281 


287 


293 


9 


4 


20 


253 


259 


265 


272 


278 


284 


290 


296 


302 


309 




21 


266 


272 


279 


285 


292 


298 


305 


311 


318 


324 


i 


1 


22 


278 


2S5 


292 


299 


306 


312 


319 


326 


333 


340 


2 


2 


23 


291 


298 


305 


312 


319 


327 


334 


341 


348 


355 


3 


2 


21 


304 


311 


319 


326 


333 


341 


348 


356 


363 


370 


4 


3 


25 


—316 


—324 


—332 


—340 


-347 


—355 


—363 


—370 


—378 


—386 


5 


4 


26 


329 


337 


345 


353 


361 


369 


377 


385 


393 


401 


6 


5 


27 


342 


350 


358 


367 


375 


383 


392 


400 


408 


417 


7 


5 


28 


354 


363 


372 


380 


389 


398 


406 


415 


423 


432 


8 


6 


29 


367 


376 


385 


394 


403 


412 


421 


430 


439 


448 


9 


7 


30 


380 


389 


398 


407 


417 


426 


435 


444 


454 


463 




31 


392 


402 


411 


421 


431 


440 


450 


459 


469 


478 


i 


1 


32 


405 


415 


425 


435 


444 


454 


464 


474 


484 


494 


2 


2 


33 


418 


428 


438 


448 


458 


469 


479 


489 


499 


509 


3 


3 


34 


430 


441 


451 


462 


472 


483 


493 


504 


514 


525 


4 


4 


35 


—443 


—454 


—465 


—475 


-486 


—497 


—508 


—519 


—529 


—540 


5 


5 


36 


456 


467 


478 


489 


500 


511 


522 


533 


544 


556 


6 


6 


37 


468 


480 


491 


502 


514 


525 


537 


548 


560 


671 


7 


8 


38 


481 


493 


504 


516 


528 


540 


551 


563 


575 


586 


8 


9 


39 


494 


506 


518 


530 


542 


554 


566 


578 


590 


602 


9 


10 


10 


506 


519 


531 


543 


556 


568 


580 


593 


605 


617 




41 


519 


531 


544 


557 


569 


582 


595 


607 


620 


633 


i 


1 


42 


531 


544 


557 


570 


583 


596 


609 


622 


635 


648 


2 


3 


43 


544 


557 


571 


584 


597 


610 


624 


637 


650 


664 


3 


4 


44 


557 


570 


584 


598 


611 


625 


638 


652 


665 


679 


4 


6 


45 


—569 


—583 


—597 


—611 


—625 


—639 


—653 


—667 


—681 


—694 


5 


7 


46 


582 


596 


610 


* 625 


639 


653 


667 


681 


696 


710 


6 


8 


47 


595 


609 


624 


638 


653 


667 


682 


696 


711 


725 


7 


10 


48 


607 


622 


637 


652 


667 


681 


696 


711 


726 


741 


8 


11 


49 


620 


635 


650 


6G5 


681 


691) 


710 


726 


741 


756 


9 


13 


50 


633 


648 


664 


679 


694 


710 


725 


741 


756 


772 




41 


42 


43 


44 


45 


46 


47 


48 


49 


50 


. i 


.2 


•3 


•4 


■5 


.6 


-7 


.8 


•9 


Correct 
tenths ii 


ons for 


1 


3 


4 


6 


7 


8 


10 


11 


13 


1 \V1 


dth. 



68o 



SURVEYING. 



TABLE X. — Continued. 
Volume of the Prismoidal Formula. 



to 

.a 










Heights. 








Corrections 


•v 




















4 


"or tenths 




















J 


'0 


41 


42 


43 


44 


45 


46 


47 


48 


49 


50 in height. 


51 


645 


661 


677 


693 


708 


724 


740 


756 


771 


787 | 


.i 


2 


52 


658 


674 


690 


706 


722 


738 


754 


770 


786 


802 | 


.2 


3 


53 


671 


687 


703 


720 


736 


752 


768 


785 


802 


818 


.3 


5 


54 


683 


700 


717 


733 


750 


767 


783 


800 


817 


833 


•4 


7 


55 


—696 


—713 


—730 


—747 


—764 


—781 


—798 


—815 


-832 


—849 


. c 


8 


56 


709 


726 


743 


760 


778 


795 


812 


830 


847 


864 


.6 


10 


57 


721 


739 


756 


774 


792 


809 


827 


844 


862 


880 


7 


12 


58 


734 


752 


770 


788 


806 


823 


841 


859 


877 


895 


8 


14 


59 


747 


765 


783 


801 


819 


833 


856 


874 


892 


910 


9 


15 


60 


759 


778 


796 


815 


833 


852 


870 


889 


907 


926 




61 


772 


791 


810 


828 


847 


866 


8S5 


991 


923 


941 


i 


2 


62 


785 


804 


823 


842 


861 


880 


899 


919 


938 


957 


2 


4 


63 


797 


817 


836 


856 


875 


894 


914 


933 


953 


972 


3 


6 


64 


810 


830 


849 


869 


889 


909 


928 


948 


968 


■ 988 


4 


8 


65 


—823 


—843 


—863 


—883 


—903 


—923 


-943 


—963 


—983 


—1003 


5 


10 


66 


835 


856 


876 


896 


917 


937 


957 


978 


998 


1019 


6 


12 


67 


848 


86!) 


889 


910 


931 


951 


972 


993 


1013 


1034 


7 


14 


68 


860 


SSI 


902 


923 


944 


965 


986 


1007 


1028 


1049 


8 


16 


69 


873 


894 


916 


937 


958 


980 


1001 


1022 


1044 


1065 


9 


18 


70 


886 


907 


929 


951 


972 


994 


1015 


1137 


1059 


1080 




71 


898 


920 


942 


964 


9S6 


1008 


1030 


1052 


1074 


1096 


i 


2 


72 


911 


933 


956 


978 


1000 


1022 


1044 


1067 


10S9 


1111 


2 


3 


73 


924 


946 


969 


991 


1014 


10: J ,6 


1059 


1081 


1104 


1127 


3 


7 


74 


936 


959 


982 


1005 


10-<l8 


1051 


1073 


1096 


1119 


1142 


4 


9 


75 


—949 


—972 


-995 


—1019 


—1042 


—1065 


—1088 


—1111 


—1134 


—1157 


5 


12 


76 


962 


985 


1009 


1032 


1056 


1079 


1102 


1126 


1149 


1173 


6 


14 


77 


974 


998 


1022 


1046 


1069 


1093 


1117 


1141 


1165 


1188 


7 


16 


78 


987 


1011 


1035 


1059 


1083 


1107 


1131 


1156 


1180 


1204 


8 


19 


79 


1000 


1024 


1048 


1073 


1097 


1122 


1146 


1170 


1195 


1219 


9 


21 


80 


1012 


1037 


1062 


1086 


1111 


1136 


1160 


1185 


1210 


1235 




81 


1025 


1050 


1075 


1100 


1125 


1150 


1175 


1200 


1225 


1250 


i 


3 


82 


1038 


1063 


1088 


1114 


1139 


1164 


1190 


1215 


1240 


1265 


2 


5 


83 


1050 


1076 


1102 


1127 


1153 


1178 


1204 


1230 


1255 


1281 


3 


8 


84 


1063 


1089 


1115 


1141 


1167 


1193 


1219 


1244 


1270 


1296 


4 


10 


85 


—1076 


—1102 


—1128 


—1154 


—1181 


—1207 


—1233 


—1259 


—1285 


—1312 


5 


13 


86 


1088 


1115 


1141 


1168 


1194 


1221 


1248 


1274 


1301 


1327 


6 


16 


87 


1101 


1128 


1155 


1181 


1208 


1235 


1262 


1289 


1316 


1343 


7 


18 


88 


1114 


1141 


1168 


1195 


1222 


1249 


1277 


1304 


1331 


1358 


8 


21 


89 


1126 


1154 


1181 


1209 


1236 


1264 


1291 


1319 


1346 


1373 


9 


24 


90 


1139 


1167 


1194 


1222 


1250 


1278 


1306 


1333 


1361 


1389 




91 


1152 


1180 


1208 


1236 


1264 


1292 


1320 


1348 


1376 


1404 


i 


3 


92 


1164 


1193 


1221 


1249 


1278 


1306 


1335 


1363 


1391 


1420 


2 


6 


93 


1177 


1206 


1234 


1263 


1292 


1320 


1349 


1378 


1406 


1435 


3 


9 


94 


1190 


1219 


1248 


1277 


1306 


1335 


1364 


1393 


1422 


1451 


4 


12 


95 


—1202 


—1231 


—1261 


—1290 


—1319 


—1349 


—1378 


—1407 


—1437 


—1466 


5 


15 


96 


1215 


1214 


1274 


1304 


1333 


1363 


1393 


1422 


1452 


1481 


6 


18 


97 


1227 


1257 


1287 


1317 


1347 


1377 


1407 


1437 


1467 


1497 


7 


21 


98 


1240 


1270 


1301 


1331 


1361 


1391 


1422 


1452 


1482 


1512 


8 


23 


99 


1253 


1283 


1314 


1344 


1375 


1406 


1436 


1467 


1497 


1528 


9 


26 


100 


1265 


1296 


1327 


1358 


1389 


1420 


1451 


1481 


1512 


1543 






41 


42 


43 


44 


45 


46 


47 


48 


49 


50 


. i 


.2 

3 


■3 


4 


•5 


.6 


•7 


.8 


•9 


Correct 
tenths i 


ions for 


1 


4 


6 


7 


8 


10 


11 


13 


a wj 


dth. 



TABLES. 



68 1 



\ 










- 














r 
^ 




Lri 


-u 


4* 


4* 


4>- 


4^ 


OJ 


OJ 


OJ 


OJ 


OJ 




— . 




O 


oc 


CN 


4^ 


to 


o 


00 


CN 


4x 


to 


O 




— 




O 


) 


o 


O 


O 


o 


o 


o 


o 


O 


o 




n 




























r 

2 


CN 


ON 


On 


On 


On 


ON 


ON 


CN 


On 


CN 


CN 


c 


m 


ora 


o 


NO 


no 


NO 


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LR3Ap'27 



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LIBRARY OF CONGRESS 



020 366 108 3 





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